Constitutive equations of linear viscoelasticity

There are various constitutive models for linear viscoelastic materials. Those models include the Maxwell model, the Kelvin-Voigt model, and the standard linear viscoelastic solid model that combines the above two models. All these models decompose stress and deformation into two summands, one that represents elastic effects and another that represents viscous effects, these models being interpretable in terms of springs and dampers. Each of these models differs in the arrangement of the springs and shock absorbers.

Another interesting property is that the constitutive equations can also be interpreted in terms of electrical circuits, in which the mechanical stress would be the equivalent of the voltage and the deformation rate would be equivalent to the current intensity. The elastic modulus would be equivalent to the capacitance of the circuit (which measures the energy storage capacity) and the viscosity to the resistance of the circuit (which measures the ability to dissipate energy).

Maxwell model

The Maxwell model is a particular case of the expression () in which , also called "long memory" viscoelastic material. A virtue of Maxwell's model is that it admits an intuitive representation in terms of springs and dissipators (dampers), more specifically Maxwell's model represents an elastic element arranged in series with a damper. The constitutive equation of the model is given by the following first order differential equation:.

This model predicts that in a material placed under constant deformation, the stresses will gradually relax until they become zero. It also predicts that if the material is placed under constant tension, the deformation will have two components, a first elastic component that appears instantly and a long-term viscous deformation that will grow over time as long as forces continue to exist. Alternatively, the constitutive equation of this model can be written as:.

By integrating the last term by parts, it is possible to rewrite the previous expression in terms of the relaxation function and the deformation rate:

To obtain the creep function, we integrate directly (), and integrating by parts we obtain:

And it is not possible to find an ordinary function to express the above in the form ().

Maxwell's model predicts that stress will decay exponentially with time in a polymer subjected to constant deformation, which fits fairly well with what has been observed experimentally for many polymers. However, an important limitation is that it does not predict the creep behavior of many polymers very reliably since in this case it predicts a linear increase in strain with time if the stress is constant, however most polymers show a decreasing strain rate with time. The main applications of this model are the modeling of thermoplastic polymers near their melting temperature, that of fresh concrete and that of numerous metals near their melting point.

Voigt-Kelvin model

The Kelvin-Voigt model or Voigt model is a particular case of the expression () in which , also called "short memory" viscoelastic material. Like the previous model, it allows a simple representation in terms of springs and dampers: the model is representable by a Newtonian damper and a spring that follows Hooke's law connected in parallel to the damper, as shown in the figure. The constitutive equation of the model can be expressed as a first-order differential equation:.

This model represents a solid that undergoes reversible viscoplastic deformation. Under the application of a constant stress the material deforms at a progressively slower rate, asymptotically reaching a quasi-steady state. When the external stress-generating forces are removed, the material relaxes to its original undeformed state. In a situation of constant stress (creep), the model is quite realistic and predicts deformations that tend to the limit σ/E for long times.

In this case the relaxation function is given by the Dirac delta (hence the name "short memory") since:.

To obtain the creep function, we look for the Laplace transform of () from which it is easy to isolate the transform of the deformation and by applying the inverse transform it is easy to arrive at:

Integrating this last expression by parts, we arrive at an expression that contains the creep function:

This model is used to explain the creep behavior of polymers. Although, like the Maxwell model, the Kelvin-Voigt model has empirical limitations. Although it models creep very well with respect to relaxation, the model generally fits less well with the behavior of viscoelastic materials. The main applications of the model are the modeling of organic polymers, rubber, rubber and wood when the load is not very high.

Standard Viscoelastic Solid Model

The last two models presented have somewhat complementary limitations. For this reason, it is considered that a model that combines characteristics of both can be a reasonable model of a viscoelastic solid. The constitutive equation of this model is given by the following differential equation:.

Using Laplace transforms, solving the voltage transform and antitransforming, a Volterra type equation is obtained as in the previous cases, which has the form:.

An integration by parts of the last term allows us to write in terms of the relaxation function and the strain rate:.

Burgers Model

The Burgers model combines features of the Maxwell and Kevin-Voigt models. The constitutive equation can be written compactly as a second-order differential equation:

Where, using the Laplace transform, the previous equation can be integrated mechanically (assuming that initially both the stress and the deformation and their rates of change are zero):

From which the relaxation function is obtained, integrating by parts:

And also through the Laplace transform the expression is obtained:.

Where:.

Which, integrated by parts, allows us to obtain the creep function:

Wiechert model

The Wiechert or Maxwell-Wiechert model (sometimes also generalized Maxwell model) is a generalization of the standard viscolastic model (and therefore also of the simple Maxwell model). This model takes into account that the relationship does not occur according to a single characteristic rhythm, but according to a distribution of time scales. This happens because there are molecular segments of different lengths, with the shorter ones contributing less than the longer ones, hence the diversity of time scales. The Weichert model models this using as many spring-damper assemblies as necessary to approximate the time scale distribution. The figure on the right represents a possible Wiechert model.[3][4][5] Given a Maxwell-Wiecher model with elements whose longitudinal moduli of elasticity are with , and whose viscosities are with , and with relaxation times given by , then the general form of the Maxwell-Wiechert model is given by:.

Starting as in the previous cases from an undeformed natural state and using the formalism of the Laplace transform, we have that the tension is expressed as:.

By doing integration by parts we arrive at:

Where the relation function directly has the form of a Prony series.

This model finds application in metals and metal alloys at temperatures lower than a quarter of their absolute melting temperature (expressed in K).

Constitutive equations of linear viscoelasticity

There are various constitutive models for linear viscoelastic materials. Those models include the Maxwell model, the Kelvin-Voigt model, and the standard linear viscoelastic solid model that combines the above two models. All these models decompose stress and deformation into two summands, one that represents elastic effects and another that represents viscous effects, these models being interpretable in terms of springs and dampers. Each of these models differs in the arrangement of the springs and shock absorbers.

Another interesting property is that the constitutive equations can also be interpreted in terms of electrical circuits, in which the mechanical stress would be the equivalent of the voltage and the deformation rate would be equivalent to the current intensity. The elastic modulus would be equivalent to the capacitance of the circuit (which measures the energy storage capacity) and the viscosity to the resistance of the circuit (which measures the ability to dissipate energy).

Maxwell model

The Maxwell model is a particular case of the expression () in which , also called "long memory" viscoelastic material. A virtue of Maxwell's model is that it admits an intuitive representation in terms of springs and dissipators (dampers), more specifically Maxwell's model represents an elastic element arranged in series with a damper. The constitutive equation of the model is given by the following first order differential equation:.

This model predicts that in a material placed under constant deformation, the stresses will gradually relax until they become zero. It also predicts that if the material is placed under constant tension, the deformation will have two components, a first elastic component that appears instantly and a long-term viscous deformation that will grow over time as long as forces continue to exist. Alternatively, the constitutive equation of this model can be written as:.

By integrating the last term by parts, it is possible to rewrite the previous expression in terms of the relaxation function and the deformation rate:

To obtain the creep function, we integrate directly (), and integrating by parts we obtain:

And it is not possible to find an ordinary function to express the above in the form ().

Maxwell's model predicts that stress will decay exponentially with time in a polymer subjected to constant deformation, which fits fairly well with what has been observed experimentally for many polymers. However, an important limitation is that it does not predict the creep behavior of many polymers very reliably since in this case it predicts a linear increase in strain with time if the stress is constant, however most polymers show a decreasing strain rate with time. The main applications of this model are the modeling of thermoplastic polymers near their melting temperature, that of fresh concrete and that of numerous metals near their melting point.

Voigt-Kelvin model

The Kelvin-Voigt model or Voigt model is a particular case of the expression () in which , also called "short memory" viscoelastic material. Like the previous model, it allows a simple representation in terms of springs and dampers: the model is representable by a Newtonian damper and a spring that follows Hooke's law connected in parallel to the damper, as shown in the figure. The constitutive equation of the model can be expressed as a first-order differential equation:.

This model represents a solid that undergoes reversible viscoplastic deformation. Under the application of a constant stress the material deforms at a progressively slower rate, asymptotically reaching a quasi-steady state. When the external stress-generating forces are removed, the material relaxes to its original undeformed state. In a situation of constant stress (creep), the model is quite realistic and predicts deformations that tend to the limit σ/E for long times.

In this case the relaxation function is given by the Dirac delta (hence the name "short memory") since:.

To obtain the creep function, we look for the Laplace transform of () from which it is easy to isolate the transform of the deformation and by applying the inverse transform it is easy to arrive at:

Integrating this last expression by parts, we arrive at an expression that contains the creep function:

This model is used to explain the creep behavior of polymers. Although, like the Maxwell model, the Kelvin-Voigt model has empirical limitations. Although it models creep very well with respect to relaxation, the model generally fits less well with the behavior of viscoelastic materials. The main applications of the model are the modeling of organic polymers, rubber, rubber and wood when the load is not very high.

Standard Viscoelastic Solid Model

The last two models presented have somewhat complementary limitations. For this reason, it is considered that a model that combines characteristics of both can be a reasonable model of a viscoelastic solid. The constitutive equation of this model is given by the following differential equation:.

Using Laplace transforms, solving the voltage transform and antitransforming, a Volterra type equation is obtained as in the previous cases, which has the form:.

An integration by parts of the last term allows us to write in terms of the relaxation function and the strain rate:.

Burgers Model

The Burgers model combines features of the Maxwell and Kevin-Voigt models. The constitutive equation can be written compactly as a second-order differential equation:

Where, using the Laplace transform, the previous equation can be integrated mechanically (assuming that initially both the stress and the deformation and their rates of change are zero):

From which the relaxation function is obtained, integrating by parts:

And also through the Laplace transform the expression is obtained:.

Where:.

Which, integrated by parts, allows us to obtain the creep function:

Wiechert model

The Wiechert or Maxwell-Wiechert model (sometimes also generalized Maxwell model) is a generalization of the standard viscolastic model (and therefore also of the simple Maxwell model). This model takes into account that the relationship does not occur according to a single characteristic rhythm, but according to a distribution of time scales. This happens because there are molecular segments of different lengths, with the shorter ones contributing less than the longer ones, hence the diversity of time scales. The Weichert model models this using as many spring-damper assemblies as necessary to approximate the time scale distribution. The figure on the right represents a possible Wiechert model.[3][4][5] Given a Maxwell-Wiecher model with elements whose longitudinal moduli of elasticity are with , and whose viscosities are with , and with relaxation times given by , then the general form of the Maxwell-Wiechert model is given by:.

Starting as in the previous cases from an undeformed natural state and using the formalism of the Laplace transform, we have that the tension is expressed as:.

By doing integration by parts we arrive at:

Where the relation function directly has the form of a Prony series.

This model finds application in metals and metal alloys at temperatures lower than a quarter of their absolute melting temperature (expressed in K).