Modified Munson-Dawson viscoplastic model

This problem considers a squared medium subject to external stress leading to creep. The material deforms following the modified Munson-Dawson's viscoplastic constitutive model described in Azabou et al. (2021).

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. The uniaxial stress is held constant at 10 MPa for a duration of 15 days before dropping to 5 MPa for another 15 days to simulate a multi-stage creep test.

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

Solutions

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

$var element = document.getElementById("moose-equation-fb6977b0-3aec-49e4-adf9-120ee43834b8");katex.render(" \\dot\\gamma_{vp} = \\begin{cases} A \\left( 1 - \\frac{\\gamma_{vp}}{\\bar{\\gamma}_{vp}} \\right)^n R(\\sigma, T) & \\text{if } \\gamma_{vp} \\leq \\bar{\\gamma}_{vp} \\\\ -B \\left( \\frac{\\gamma_{vp}}{\\bar{\\gamma}_{vp}} - 1 \\right)^m R(\\sigma, T) & \\text{if } \\gamma_{vp} \\geq \\bar{\\gamma}_{vp} \\end{cases}", element, {displayMode:true,throwOnError:false});$

where, $var element = document.getElementById("moose-equation-75c94b97-ec53-471c-868b-72f2196fa36e");katex.render("A", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-4793db96-6d93-42a5-bb7a-a7e12d4b4e47");katex.render("B", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-a7a614c1-2368-40c3-901e-e837c248e518");katex.render("m", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-76a82f8f-0df6-4a74-8b23-792d0a9fc81e");katex.render("n", element, {displayMode:false,throwOnError:false});$ are material parameters, with $var element = document.getElementById("moose-equation-efca326a-1e7e-469d-8902-e9e3fcbb6bd7");katex.render("A > 0", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-271f4307-222c-4a0d-b851-1ada0df4243c");katex.render("B > 0", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-9836b5c0-0e15-45e0-b5ba-d448d36b10ea");katex.render("m > 1", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-0672b0b6-dc13-46f6-81e7-953e71172a9d");katex.render("n > 1", element, {displayMode:false,throwOnError:false});$ Azabou et al. (2021). The parameter $var element = document.getElementById("moose-equation-45591d2b-e444-46e6-92d2-a6ca676a1831");katex.render("\\bar{\\gamma}_{vp}", element, {displayMode:false,throwOnError:false});$ describes the saturation strain, similar to that expressed in the Munson-Dawson model, and corresponds to the threshold of transient deformation. It is defined as:

$(1)var element = document.getElementById("moose-equation-f7fb0f91-07bd-4d20-a140-f1cf8e21dc7c");katex.render(" \\bar{\\gamma}_{vp}(\\sigma, T)= \\left\\langle \\frac{q}{A_1} \\right\\rangle^{n_1} ", element, {displayMode:true,throwOnError:false});$ here $var element = document.getElementById("moose-equation-b4bd9d6c-16f1-4b71-b42d-c8a749f31da7");katex.render("A_1", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-c79230d1-2d98-4e17-8e9d-2b47eb87f256");katex.render("n_1", element, {displayMode:false,throwOnError:false});$ are material parameters and $var element = document.getElementById("moose-equation-2b40f5c2-ad48-4439-a9c0-4f27dcce0023");katex.render("q = \\sqrt{\\frac{3}{2}\\tau_{ij}\\tau_{ij}}", element, {displayMode:false,throwOnError:false});$ is the equivalent stress, representing the loading function. $var element = document.getElementById("moose-equation-0520e1e3-bab2-44fa-99c7-ed733cb74d09");katex.render("\\tau_{ij}", element, {displayMode:false,throwOnError:false});$ is the deviatoric stress tensor.

The function $var element = document.getElementById("moose-equation-7265b2b2-a9c2-45ae-a635-c79ab89572b0");katex.render("R", element, {displayMode:false,throwOnError:false});$ corresponds to the Lemaitre’s scalar equivalent creep strain evolution, ensuring that $var element = document.getElementById("moose-equation-9a48595e-7c8c-4eac-a52a-c4162f9c1986");katex.render("\\dot\\gamma_{vp}", element, {displayMode:false,throwOnError:false});$ have similar order of magnitude as the Lemaitre's scalar equivalent creep model Azabou et al. (2021): $var element = document.getElementById("moose-equation-f0bb0d6f-9e04-4b69-bec0-1e024fb6f2e2");katex.render(" R(\\sigma, T) = \\exp \\left( A_R \\left( \\frac{1}{T_r} - \\frac{1}{T} \\right) \\right) \\left\\langle \\frac{q}{A_2} \\right\\rangle^{n_2}", element, {displayMode:true,throwOnError:false});$

This model is equivalent to the Munson-Dawson model provided the following conditions are satisfied:

Parameter $var element = document.getElementById("moose-equation-29a07379-2990-4fe8-afb8-b99742fae0cc");katex.render("A = \\exp{\\Delta}", element, {displayMode:false,throwOnError:false});$
Parameter $var element = document.getElementById("moose-equation-738ed82e-0e87-4747-b771-dd2741656750");katex.render("n = \\begin{cases} -0.0098\\Delta^3 + 0.2040\\Delta^2 + 0.5622\\Delta + 2.0252, & \\text{if } \\Delta \\leq 6 \\\\ 1.9987\\Delta - 1.4567, & \\text{if } \\Delta > 6 \\end{cases}", element, {displayMode:false,throwOnError:false});$
Parameter $var element = document.getElementById("moose-equation-ebb5e0b8-a743-4b30-8e73-70aee1c3e59e");katex.render("B = 0", element, {displayMode:false,throwOnError:false});$ (i.e., the transient strain threshold is not exceeded). Hence, the strain-rate is always transient.

The analytical solution for this problem is given as Azabou et al. (2021):

$var element = document.getElementById("moose-equation-d22c756c-cb2a-419b-a62f-0954063e96f9");katex.render(" \\gamma_{vp} \\left( t \\right) = {\\gamma_{vp}}_0 + (\\bar{\\gamma}_{vp} + {\\gamma_{vp}}_0) \\, \\xi\\left(t\\right),", element, {displayMode:true,throwOnError:false});$ $var element = document.getElementById("moose-equation-68ce718a-a933-48d6-b79e-503efb4c8b9f");katex.render(" \\xi \\left( t \\right) = \\left(1 - \\frac{1}{1 + V(t - t_0)} \\right)^{\\frac{1}{k-1}},", element, {displayMode:true,throwOnError:false});$ $var element = document.getElementById("moose-equation-0c0c19f2-8c4c-4809-9556-57453fcb8f88");katex.render(" V = (k - 1) \\, U \\, \\frac{A_{1}^{n_{1}}}{A_{2}^{n_{2}}} \\, q^{n_{2} - n_{1}},", element, {displayMode:true,throwOnError:false});$ $var element = document.getElementById("moose-equation-4aac88d3-b21f-424c-b98c-6ba167c44269");katex.render(" \\begin{cases} \\text{when } \\gamma_{vp} \\leq \\bar{\\gamma}_{vp}; \\quad k = n; \\quad U = A \\left( 1 - \\frac{{\\gamma_{vp}}_0}{\\bar{\\gamma}_{vp}} \\right)^{n-1} \\\\ \\text{when } \\gamma_{vp} \\geq \\bar{\\gamma}_{vp}; \\quad k = m; \\quad U = B \\left( \\frac{{\\gamma_{vp}}_0}{\\bar{\\gamma}_{vp}} - 1 \\right)^{m-1} \\end{cases}", element, {displayMode:true,throwOnError:false});$

The following creep curve shows a comparison between the analytical and the numerical solutions of the modified Munson-Dawson model.

Figure 2: Creep strain evolution in a Munson-Dawson's viscoplastic medium.

Complete Source Files

munson_dawson.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"
}

Lemaitre s viscopl. . .

Setup
Solutions
Complete Source Files
References