Lemaitre's viscoplastic model

This problem considers a squared medium subject to external stress leading to creep. The material deforms following the Lemaitre's viscoplastic constitutive model.

Setup

The squared medium is subject to a compressive horizontal uniaxial stress at a constant temperature, resulting in a uniaxial deformation. The setup is sketched in Figure 1.

Figure 1: Setup for the Lemaitre's viscoplastic medium.

Solutions

The scalar equivalent creep strain evolution is governed by the following constitutive model:

var element = document.getElementById("moose-equation-db736d98-0296-4a11-aac0-7e42d3e2c9c9");katex.render("\\dot{\\gamma}_{vp} = A \\left( \\frac{q}{A_2} \\right)^{\\frac{\\beta}{\\alpha}} \\gamma_{vp}^{1-\\frac{1}{a}},", element, {displayMode:true,throwOnError:false});

where $var element = document.getElementById("moose-equation-f77ebca4-a5de-4b40-9da0-261d86e94802");katex.render("A", element, {displayMode:false,throwOnError:false});$ is a time-scaling model parameter unique to the material under consideration. Azabou et al. (2021) expressed this parameter in terms of the Arrhenius law as $var element = document.getElementById("moose-equation-e4f960c2-655b-4d1f-bcec-f54cff20a068");katex.render("\\alpha \\exp\\left(\\frac{1}{T_r}-\\frac{1}{T}\\right)", element, {displayMode:false,throwOnError:false});$ to describe the behavior of rock salt. In this case, $var element = document.getElementById("moose-equation-7542e464-1ded-411e-9d9a-e26252cec3e3");katex.render("A \\approx \\alpha", element, {displayMode:false,throwOnError:false});$ for isothermal conditions. $var element = document.getElementById("moose-equation-d341413f-c69c-479e-a49e-7c30b1454c5a");katex.render("\\alpha", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-54106fd6-d94e-49f1-830b-eb7a1f2bf905");katex.render("\\beta", element, {displayMode:false,throwOnError:false});$ are material parameters. $var element = document.getElementById("moose-equation-f33bef2d-e919-400f-88bb-646c253fa3ca");katex.render("A_2", element, {displayMode:false,throwOnError:false});$ is a normalizing parameter always taken as 1.0. $var element = document.getElementById("moose-equation-f7dc5f56-ce73-4985-a461-3b63561477f1");katex.render("\\gamma_{vp}", element, {displayMode:false,throwOnError:false});$ is the scalar equivalent creep strain.

The analytical solution for this problem is given as:

var element = document.getElementById("moose-equation-de5ed33a-c559-4a92-b4a9-1951639ecf3b");katex.render("\\gamma_{vp} = \\left( \\frac{A}{\\alpha} \\right)^{\\alpha} \\left[ \\left( \\frac{q}{A_2} \\right)^{\\beta} \\right] t^{a},", element, {displayMode:true,throwOnError:false});

where $var element = document.getElementById("moose-equation-95e69d4e-ee45-467a-a042-40bbab2b9ecd");katex.render("t", element, {displayMode:false,throwOnError:false});$ is time.

The following creep curve shows a comparison between the analytical and the numerical solutions of the Lemaitre model.

Figure 2: Creep strain evolution in a Lemaitre's viscoplastic medium.

Complete Source Files

lemaitre.i

References

M Azabou, Ahmed Rouabhi, L Blanco-Martín, F Hadj-Hassen, M Karimi-Jafari, and G Hévin. Rock salt behavior: from laboratory experiments to pertinent long-term predictions. International Journal of Rock Mechanics and Mining Sciences, 142:104588, 2021.

@article{azabou2021rock,
    author = "Azabou, M and Rouabhi, Ahmed and Blanco-Mart{\'\i}n, L and Hadj-Hassen, F and Karimi-Jafari, M and H{\'e}vin, G",
    title = "Rock salt behavior: From laboratory experiments to pertinent long-term predictions",
    journal = "International Journal of Rock Mechanics and Mining Sciences",
    volume = "142",
    pages = "104588",
    year = "2021",
    publisher = "Elsevier"
}

Modified Munson Da. . .

Setup
Solutions
Complete Source Files
References