Maxwell model#Effect of a sudden deformation

Image:Maxwell diagram.svg

The Maxwell model is represented by a purely viscous damper and a purely elastic spring connected in series,{{cite book |doi=10.1016/B978-0-12-174252-2.50005-3 |chapter=Viscoelastic Stress Strain Constitutive Relations |title=Theory of Viscoelasticity |date=1982 |last1=Christensen |first1=R.M. |pages=1–34 |isbn=978-0-12-174252-2 }} as shown in the diagram. If, instead, we connect these two elements in parallel, we get the generalized model of a solid Kelvin–Voigt material.

In Maxwell configuration, under an applied axial stress, the total stress, $\backslash sigma\_\backslash mathrm\{Total\}$ and the total strain, $\backslash varepsilon\_\backslash mathrm\{Total\}$ can be defined as follows:

:$\backslash sigma\_\backslash mathrm\{Total\}=\backslash sigma\_\{\backslash rm\; D\}\; =\; \backslash sigma\_\{\backslash rm\; S\}$

:$\backslash varepsilon\_\backslash mathrm\{Total\}=\backslash varepsilon\_\{\backslash rm\; D\}+\backslash varepsilon\_\{\backslash rm\; S\; \}$

where the subscript D indicates the stress–strain in the damper and the subscript S indicates the stress–strain in the spring. Taking the derivative of strain with respect to time, we obtain:

:$\backslash frac\; \{d\backslash varepsilon\_\backslash mathrm\{Total\}\}\; \{dt\}\; =\; \backslash frac\; \{d\backslash varepsilon\_\{\backslash rm\; D\}\}\; \{dt\}\; +\; \backslash frac\; \{d\backslash varepsilon\_\{\backslash rm\; S\}\}\; \{dt\}\; =\; \backslash frac\; \{\backslash sigma\}\; \{\backslash eta\}\; +\; \backslash frac\; \{1\}\; \{E\}\; \backslash frac\; \{d\backslash sigma\}\; \{dt\}$

where E is the elastic modulus and η is the material coefficient of viscosity. This model describes the damper as a Newtonian fluid and models the spring with Hooke's law.

In a Maxwell material, stress σ, strain ε and their rates of change with respect to time t are governed by equations of the form:

:$\backslash frac\; \{1\}\; \{E\}\; \backslash frac\; \{d\backslash sigma\}\; \{dt\}\; +\; \backslash frac\; \{\backslash sigma\}\; \{\backslash eta\}\; =\; \backslash frac\; \{d\backslash varepsilon\}\; \{dt\}$

or, in dot notation:

:$\backslash frac\; \{\backslash dot\; \{\backslash sigma\}\}\; \{E\}\; +\; \backslash frac\; \{\backslash sigma\}\; \{\backslash eta\}=\; \backslash dot\; \{\backslash varepsilon\}$

The equation can be applied either to the shear stress or to the uniform tension in a material. In the former case, the viscosity corresponds to that for a Newtonian fluid. In the latter case, it has a slightly different meaning relating stress and rate of strain.

The model is usually applied to the case of small deformations. For the large deformations we should include some geometrical non-linearity. For the simplest way of generalizing the Maxwell model, refer to the upper-convected Maxwell model.

Image:Maxwell deformation.PNG

If a Maxwell material is suddenly deformed and held to a strain of $\backslash varepsilon\_0$, then the stress decays on a characteristic timescale of $\backslash frac\{\backslash eta\}\{E\}$, known as the relaxation time. The phenomenon is known as stress relaxation.

The picture shows dependence of dimensionless stress $\backslash frac\; \{\backslash sigma(t)\}\; \{E\backslash varepsilon\_0\}$

upon dimensionless time $\backslash frac\{E\}\{\backslash eta\}\; t$:

If we free the material at time $t\_1$, then the elastic element will spring back by the value of

:$\backslash varepsilon\_\backslash mathrm\{back\}\; =\; -\backslash frac\; \{\backslash sigma(t\_1)\}\; E\; =\; \backslash varepsilon\_0\; \backslash exp\; \backslash left(-\backslash frac\{E\}\{\backslash eta\}\; t\_1\backslash right).$

Since the viscous element would not return to its original length, the irreversible component of deformation can be simplified to the expression below:

:$\backslash varepsilon\_\backslash mathrm\{irreversible\}\; =\; \backslash varepsilon\_0\; \backslash left[1-\; \backslash exp\; \backslash left(-\backslash frac\{E\}\{\backslash eta\}\; t\_1\backslash right)\backslash right].$

If a Maxwell material is suddenly subjected to a stress $\backslash sigma\_0$, then the elastic element would suddenly deform and the viscous element would deform with a constant rate:

:$\backslash varepsilon(t)\; =\; \backslash frac\; \{\backslash sigma\_0\}\; E\; +\; t\; \backslash frac\{\backslash sigma\_0\}\; \backslash eta$

If at some time $t\_1$ we released the material, then the deformation of the elastic element would be the spring-back deformation and the deformation of the viscous element would not change:

:$\backslash varepsilon\_\backslash mathrm\{reversible\}\; =\; \backslash frac\; \{\backslash sigma\_0\}\; E,$

:$\backslash varepsilon\_\backslash mathrm\{irreversible\}\; =\; t\_1\; \backslash frac\{\backslash sigma\_0\}\; \backslash eta.$

The Maxwell model does not exhibit creep since it models strain as linear function of time.

If a small stress is applied for a sufficiently long time, then the irreversible strains become large. Thus, Maxwell material is a type of liquid.

If a Maxwell material is subject to a constant strain rate $\backslash dot\{\backslash epsilon\}$then the stress increases, reaching a constant value of

$\backslash sigma=\backslash eta\; \backslash dot\{\backslash varepsilon\}$

In general

$\backslash sigma\; (t)=\backslash eta\; \backslash dot\{\backslash varepsilon\}(1-\; e^\{-Et/\backslash eta\})$

Image:Maxwell relax spectra.PNG

The complex dynamic modulus of a Maxwell material would be:

:$E^*(\backslash omega)\; =\; \backslash frac\; 1\; \{1/E\; -\; i/(\backslash omega\; \backslash eta)\; \}\; =\; \backslash frac\; \{E\backslash eta^2\; \backslash omega^2\; +i\; \backslash omega\; E^2\backslash eta\}\; \{\backslash eta^2\; \backslash omega^2\; +\; E^2\}$

Thus, the components of the dynamic modulus are :

:$E\_1(\backslash omega)\; =\; \backslash frac\; \{E\backslash eta^2\; \backslash omega^2\; \}\; \{\backslash eta^2\; \backslash omega^2\; +\; E^2\}\; =\; \backslash frac\; \{(\backslash eta/E)^2\backslash omega^2\}\; \{(\backslash eta/E)^2\; \backslash omega^2\; +\; 1\}\; E\; =\; \backslash frac\; \{\backslash tau^2\backslash omega^2\}\; \{\backslash tau^2\; \backslash omega^2\; +\; 1\}\; E$

and

:$E\_2(\backslash omega)\; =\; \backslash frac\; \{\backslash omega\; E^2\backslash eta\}\; \{\backslash eta^2\; \backslash omega^2\; +\; E^2\}\; =\; \backslash frac\; \{(\backslash eta/E)\backslash omega\}\; \{(\backslash eta/E)^2\; \backslash omega^2\; +\; 1\}\; E\; =\; \backslash frac\; \{\backslash tau\backslash omega\}\; \{\backslash tau^2\; \backslash omega^2\; +\; 1\}\; E$

The picture shows relaxational spectrum for Maxwell material. The relaxation time constant is $\backslash tau\; \backslash equiv\; \backslash eta\; /\; E$.

• Burgers material
• Generalized Maxwell model
• Kelvin–Voigt material
• Oldroyd-B model
• Standard linear solid model
• Upper-convected Maxwell model

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Category:Non-Newtonian fluids

Category:Materials science

Category:James Clerk Maxwell