Standard linear solid model

The Standard Linear Solid Model (SLS) is a method of modeling the behavior of a viscoelastic material using a linear combination of springs and dashpots that represent elastic and viscous components, respectively. Previously, the Maxwell model and the Kelvin-Voight model were used. These models proved insufficient, however; Maxwell's model was inaccurate at predicting creep, and Kelvin-Voight's model, while good at predicting creep, was less accurate when used to model relaxation. By matching derived theoretical relationships to experimental data, the SLS was created.

In contrast to Maxwell's and Kelvin and Voight's model, the SLS is slightly more complex, involving elements both in series and in parallel. Springs, which represent the elastic component of a viscoelastic material, obey Hooke's Law:

Failed to parse (syntax error): {\displaystyle δ = Eε}

where δ is the applied stress, and ε is the strain.

Dashpots represent the viscous component of a viscoelastic material. In these elements, the applied stress varies with the time rate of change of the strain:

Failed to parse (syntax error): {\displaystyle δ = η \frac{dε}{dt} }

where η is the elasticity of the viscous component.

These elements are connected as shown below:

In order to model this system, the following physical relations must be realized:

For parallel components: Failed to parse (syntax error): {\displaystyle δtot = δ1 + δ2} , and Failed to parse (syntax error): {\displaystyle εtot = ε1 = ε2}

For series components: Failed to parse (syntax error): {\displaystyle δtot = δ1 = δ2} , and Failed to parse (syntax error): {\displaystyle εtot = ε1 + ε2}

Using these relationships, the system can be modeled as follows:

Failed to parse (syntax error): {\displaystyle /frac {dε}{dt} = /frac{( /frac{E2}{η} ) (/frac{η}{E2} δ + /frac {dδ}{dt} - E1ε)}{E1+E2} }