Linear Maxwell viscoelasticity

This problem considers a squared medium subject to external velocities leading to a stress build-up. The material deforms following a linear Maxwell viscoelastic constitutive model. This problem was first introduced by Gerya and Yuen (2007).

Setup

The squared medium is subject to a compressive velocity horizontally and tension velocity vertically resulting in a deviatoric stress build-up. The setup is sketched in Figure 1.

Figure 1: Setup for the linear Maxwell viscoelastic medium.

Solutions

The stress evolution is governed by the following constitutive model:

var element = document.getElementById("moose-equation-f9528adb-496f-4522-9aa1-ed1f7bf86050");katex.render("\\dot{\\tau}_{ij} = 2 G \\left(\\dot{e}_{ij} - \\frac{\\tau_{ij}}{2 \\eta}\\right)", element, {displayMode:true,throwOnError:false});

where $var element = document.getElementById("moose-equation-dc44f530-914b-463f-9160-7718ffbc345a");katex.render("\\tau_{ij}", element, {displayMode:false,throwOnError:false});$ is the deviatoric stress tensor and $var element = document.getElementById("moose-equation-7b751cab-e154-4e82-aa08-d56a59f0af4c");katex.render("e_{ij}", element, {displayMode:false,throwOnError:false});$ the deviatoric strain tensor.

The solution for this problem is given by Gerya and Yuen (2007). The stress solution is given as:

var element = document.getElementById("moose-equation-d2041e0b-a1c7-4d6c-b961-aa7f126b8177");katex.render("\\tau = \\eta \\dot{\\gamma} \\left(1 - \\exp\\left(- \\frac{G t}{\\eta}\\right)\\right)", element, {displayMode:true,throwOnError:false});

where:

var element = document.getElementById("moose-equation-f93976ac-4071-4268-9d9c-aada3956606c");katex.render("\\sigma_{xx} = - \\tau, \\quad \\sigma_{yy} = \\tau.", element, {displayMode:true,throwOnError:false});

var element = document.getElementById("moose-equation-ea26f78f-b00e-41dc-8d63-673a7909129e");katex.render("\\dot{\\gamma} = \\frac{v}{h}", element, {displayMode:true,throwOnError:false});

With the following dimensionless quantities: $var element = document.getElementById("moose-equation-63c3861d-e8d2-4f45-a819-a8442c0df087");katex.render("\\overline{t} = \\frac{G}{\\eta} t", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-bcfa11ec-f8ad-4ee0-9202-85654d96be1a");katex.render("\\overline{\\tau} = \\frac{\\tau}{\\eta \\dot{\\gamma}}", element, {displayMode:false,throwOnError:false});$ , the solution can be expressed as:

(1)var element = document.getElementById("moose-equation-2f6f7252-bbfe-4e34-baae-cb729b4fc7e5");katex.render("\\overline{\\tau} = 1 - \\exp\\left(- t\\right)", element, {displayMode:true,throwOnError:false});

Figure 2: Stress evolution in a linear viscoelastic Maxwell medium.

Complete Source Files

linear_maxwell.i

References

Taras V. Gerya and David A. Yuen. Robust characteristics method for modelling multiphase visco-elasto-plastic thermo-mechanical problems. Physics of the Earth and Planetary Interiors, 163(1-4):83–105, 2007. doi:10.1016/j.pepi.2007.04.015.

@article{Gerya2007,
    author = "Gerya, Taras V. and Yuen, David A.",
    year = "2007",
    title = "{Robust characteristics method for modelling multiphase visco-elasto-plastic thermo-mechanical problems}",
    journal = "Physics of the Earth and Planetary Interiors",
    issn = "0031-9201",
    doi = "10.1016/j.pepi.2007.04.015",
    pages = "83--105",
    number = "1-4",
    volume = "163"
}

Linear Kelvin visc. . .

Setup
Solutions
Complete Source Files
References