2. Benchmark solution

2.1. Solutions obtained for model BETON_UMLV

2.1.1. Calculation method

This section presents the complete analytical resolution of the problem of a test body subjected to a field of homogeneous and unidirectional stresses applied instantaneously at the initial moment and maintained constant thereafter (case of a creep test in simple compression):

(2.1)\[ \ mathrm {\ sigma} = {\ mathrm {\ sigma}} _ {0} {e} _ {z}\ otimes {e} _ {z}\]

Whose decomposition into spherical and deviatoric parts is written as:

(2.2)\[ \ mathrm {\ sigma} =\ underset {\ text {spherical part}} {\ underset {} {\ frac {1} {3} {\ mathrm {\ sigma}} _ {0} I}}} +\ underset {\ text {spherical part}} {\ underset {} {\ frac {2} {3}} {\ mathrm {\ sigma}} {\ mathrm {\ sigma}} {\ mathrm {\ sigma}}} _ {0} {e} _ {z}\ otimes {e} _ {e} _ {z} _ {z} -\ frac {1} {3} {\ mathrm {\ sigma}} _ {0}\ left ({e} _ {x} _ {x} _ {y}}\ left ({e} _ {y} _ {y}\ right) _ {x}\ right)}}\]

By operating a spherical/deviatoric decomposition identical to that of the stresses, axial deformation is written in the form:

(2.3)\[ {\ mathrm {\ epsilon}} _ {\ mathit {zz}}} = {\ mathrm {\ epsilon}}} ^ {\ mathit {fs}}\ left (\ frac {{\ mathrm {\ mathrm {\ sigma}}}} _ {\ sigma}}} _ {0}}} _ {0}}} {0}} {3}\ right) + {\ mathrm {\ epsilon}}} ^ {\ mathit {fd}}\ left (\ frac {2 {\ mathrm {\ sigma}}} _ {0}} {3}\ right)\]

It is therefore necessary to solve the response to a spherical stress level and to a deviatory stress level successively.

2.1.2. Solving the equations that make up spherical creep [bib2]

The process of spherical creep deformation is governed by the following system of coupled equations (equations () and (), cf. [R7.01.06]):

\[\]

: label: eq-4

{dot {mathrm {epsilon}}}}} ^ {mathit {fs}}} =frac {1} {{mathrm {eta}}} _ {r} ^ {s}}}left (h {mathrm {s}}}left (h {mathrm {epsilon}}}}left (h {mathrm {epsilon}}}}left (h {mathrm {sigma}}}} ^ {s}}left (h {mathrm {epsilon}}}}left (h {mathrm {sigma}}}} ^ {s}}left (h {mathrm {epsilon}}}}}left (h {r} ^ {mathit {fs}}right) - {dot {mathrm {epsilon}}}} _ {i} ^ {mathit {fs}} ^ {mathit {fs}}

where \({k}_{r}^{s}\) refers to the apparent stiffness associated with the skeleton formed by hydrate blocks at the mesoscopic scale and \({\mathrm{\eta }}_{r}^{s}\) the apparent viscosity associated with the diffusion mechanism within the capillary porosity.

\[\]

: label: eq-5

{dot {mathrm {epsilon}}}} _ {i}}} _ {i}} ^ {mathit {fs}} =frac {1} {{mathrm {eta}}} _ {r} ^ {s}} _ {i}} ^ {s} {mathrm {epsilon}}} _ {r} ^ {s}} _ {r} ^ {s}}} _ {r} ^ {s}} {1} _ {r} ^ {s}}} _ {r} ^ {s}} {1} _ {mathit {fs}}} -left ({k} _ {r} ^ {s} + {k} + {k} _ {k} _ {s}right) {mathrm {epsilon}} _ {i} ^ {mathit {fs}}right] -right] -left] -left [h {mathrm {sigma}}} ^ {s}}right] -left [h {mathrm {sigma}}} ^ {s} {epsilon}} _ {r} ^ {s} {s} {s} {s}right] -left] -left [h {mathrm {sigma}}} ^ {epsilon}} _ {r} ^ {s}} {mathit {fs}}}right] -left silon}} _ {r} ^ {mathit {fs}}right] ⟩} ^ {text {+}} ^ {text {+}}

where:math: {k} _ {i} ^ {s} refers to the apparent stiffness intrinsically associated with hydrates at the microscopic scale and:math: {mathrm {eta}}} _ {i} ^ {s} the apparent viscosity associated with the mechanism of interfoliar diffusion. In (), the square brackets:math: {⟨⟩}} ^ {text {+}} refer to the Mac Cauley operator: :math: {⟨x⟩}} ^ {text {+}}} =frac {+}}} =frac {1} {1} {2}left (x+|x|right)}} =frac {1} {2} {2}left (x+|x|right)

Solving the previous system of coupled equations requires distinguishing between two cases according to the sign of the quantity included in the Mac Cauley brackets. In the following, we present the analytical resolution of the response at a constraint level \({\mathrm{\sigma }}^{s}\). Relative humidity is assumed to be invariant; the medium is saturated with water.

2.1.2.1. The case of short-term creep

At the initial instant, \(t=0\), a positive \({\mathrm{\sigma }}^{s}\) spherical constraint is applied. The reversible and irreversible creep deformations are equal to zero (initial conditions). The system equation () is therefore written as:

(2.4)\[ {\ dot {\ mathrm {\ epsilon}}}} _ {i}}} _ {i}} ^ {i} ^ {i}} {{\ mathrm {\ eta}}} _ {i} ^ {s}}} _ {i} ^ {s}}} {i} ^ {s}}} _ {i} eta}}} _ {i} eta}}} _ {i} eta}}} _ {i} ^ {eta}}} _ {i} ^ {s}}} _ {i} ^ {s} 0- {s}} 0- {s}}} {\ mathrm {\ sigma}} ^ {s} ⟩}} ^ {\ text {+}}} ^ {\ text {+}}} =\ frac {1} {\ mathrm {\ eta}} _ {i} ^ {s}} {1} ^ {s}}} _ {i} ^ {s}}} _ {i} ^ {\ text {s}}}} {1}} _ {i} ^ {s}}} {\]

The rate of irreversible creep deformation is therefore equal to zero. It is deduced from this that the irreversible creep deformation is also equal to zero. The rate of irreversible deformation remains equal to zero until now

, defined by the relationship ():

(2.5)\[ 2 {k} _ {r} ^ {s} {\ mathrm {\ epsilon}}} _ {r} ^ {\ mathit {fs}} ({t} _ {0}) - {\ mathrm {\ sigma}}}} ^ {\ sigma}}} ^ {\ sigma}}} ^ {\ mathit {\ sigma}}} ^ {\ mathit {fs}}} ^ {\ mathit {fs}}} ^ {\ mathit {fs}}} ^ {\ mathit {fs}}} ^ {\ mathit {fs}}} (t} _ {0}) =\ frac {{\ mathrm {\ sigma}} ^ {s}} {2 {k} _ {r} ^ {s}}\]

Until instant \(t={t}_{0}\), reversible creep deformation is defined by the following relationship:

(2.6)\[ {\ dot {\ mathrm {\ epsilon}}}} _ {r}}} _ {r} ^ {s}} =\ frac {1} {{\ mathrm {\ eta}}} _ {r} ^ {s}}}\ left ({\ mathrm {\ epsilon}}}}\ left ({\ mathrm {\ sigma}}}}\ left ({\ mathrm {\ sigma}}}} ^ {s}} _ {r}} {\ mathrm {\ epsilon}}}\ left ({\ mathrm {\ sigma}}}} ^ {s}} _ {r}}} ^ {s}\ right)\ Rightarrow {\ mathrm {\ epsilon}}} _ {r} ^ {s} (t) =\ frac {{\ mathrm {\ sigma}} ^ {s}}} {{s}}} {{k}}} {{k} _ {r} _ {r} ^ {s}}\ left (-\ frac {t}}} {{\ mathrm {\ sigma}}} ^ {\ mathrm {\ sigma}}} ^ {\ sigma}} ^ {\ sigma}} ^ {\ sigma}} ^ {\ sigma}} ^ {\ sigma}} ^ {\ sigma}} ^ {\ sigma}} ^ {\ sigma}} ^ {\ sigma}} ^ {\ sigma}} ^ {\ sigma}} ^ {thrm {\ tau}} _ {r} ^ {s}}\ right)\ right]\]

Quantity \({\mathrm{\tau }}_{r}^{s}=\frac{{\mathrm{\eta }}_{r}^{s}}{{k}_{r}^{s}}\) is the characteristic time associated with reversible creep deformation. The moment \({t}_{0}\) is therefore defined by the relation ():

(2.7)\[ {\ mathrm {\ epsilon}} _ {r}} ^ {\ mathit {fs}} =\ frac {{\ mathrm {\ sigma}} ^ {s}} {2 {k} _ {r} _ {r}} ^ {s}} ^ {s}} =\ frac {{\ mathit {fs}}} =\ frac {{\ mathrm {fs}}} =\ frac {{\ mathrm {\ sigma}} {{s}} {{k}}} {r} _ {r} _ {r} _ {r}} ^ {s}}}\ left [1-\ mathrm {exp}\ left (-\ frac {{t} _ {0}}} {{\ mathrm {\ tau}} _ {r} ^ {s}}\ right)\ right]\ right]\ Rightarrow {t} _ {0} _ {0} =\ mathrm {\ tau}} _ {r} ^ {s}\ approx 0.0} =\ mathrm {ln} (2) {\ mathrm {\ tau}}} _ {r} ^ {s}}\ approx 0.69 {\ mathrm {\ tau}} _ {r} ^ {s}\]

Reversible and irreversible creep deformations are therefore determined by:

(2.8)\[\begin{split} \ {\ begin {array} {c} {\ mathrm {\ epsilon}}} _ {r}} ^ {\ mathit {fs}} (t) =\ frac {{\ mathrm {\ sigma}}} ^ {s}}} ^ {s}}}\ left [1-\ mathrm {exp}}\ left (-\ frac {t}}}\ left (-\ frac {t}} {\ mathrm {\ tau}} _ {r} ^ {s}} {s}}\ right)\ right)\\ {\ mathrm {\ epsilon}} _ {i} ^ {\ mathit {fs}} {\ mathit {fs}}} (t) =0\ end {array}}\end{split}\]

When calculating the creep deformations for \(t>{t}_{0}\), the new initial conditions are therefore:

(2.9)\[ \ {\ begin {array} {c} {\ mathrm {\ epsilon}}} _ {r}} ^ {\ mathit {fs}} ({t} _ {0}) =\ frac {{\ mathrm {\ mathrm {\ sigma}}}} ^ {sigma}}} ^ {s}} {\ mathrm {\ epsilon}}} _ {i} ^ {\ mathit {fs}} ({t} _ {0}) =0\ end {array}}\]

2.1.2.2. The case of long-term creep

By expressing the rates of reversible and irreversible creep deformations as a function of creep deformations, the relationship is then obtained:

(2.10)\[\begin{split} \ {\ begin {array} {c} {\ dot {\ mathrm {\ epsilon}}}} _ {r} ^ {\ mathit {fs}} =\ left (-\ frac {{k} _ {r} _ {r}} ^ {s}} {r} ^ {s}} _ {r}} ^ {s}} -4\ frac {{k} _ {r}} ^ {s}} _ {r} ^ {s}}} -4\ frac {{k} _ {r}} ^ {s}} {s}}}} {{\ mathrm {\ eta}}} _ {i} ^ {s}}}\ right) {\ mathrm {\ epsilon}} _ {r} ^ {\ mathit {fs}}} +\ left (2\ frac {{k}} _ {{k}} _ {{k}} _ {i} {k}} _ {i} ^ {s}}\ right) {\ mathrm {\ epsilon}} _ {i} ^ {\ mathit {fs}} ^ {\ mathit {fs}}} +\ left (\ frac {1} {\ mathrm {\ eta}}} _ {r} ^ {s}}} _ {r} ^ {s}}} _ {r} ^ {s}}} _ {r} ^ {s}}} _ {r} ^ {s}}}} +\ frac {2}} ^ {s}}} +\ frac {2}} ^ {s}}} +\ frac {2}} ^ {s}}} +\ frac {2}} ^ {s}}} +\ frac {2}} ^ {s}}} +\ frac {2}} ^ {s}}} ^ {s}\\ {\ dot {\ mathrm {\ epsilon}}}}} _ {i}} ^ {\ mathit {fs}} =\ left (2\ frac {{k} _ {r} ^ {s}}}} {{\ mathrm {\ epsilon}}}}} {{s}}}\ right) {\ mathrm {\ epsilon}}} _ {r} {s}}} {\ mathrm {\ epsilon}}} {{s}}} {\ mathrm {\ epsilon}}} {{s}}} {{r}}}} {\ mathrm {\ epsilon}}} {\ mathrm {\ epsilon}}}} {{r}}}} {{r}} {\ mathit {fs}} +\ left (-\ frac {{k} _ { i} ^ {s}} {{\ mathrm {\ eta}}} _ {i} ^ {s}}\ right) {\ mathrm {\ epsilon}} _ {i} ^ {\ mathit {fs}}} +\ left (-\ left (-\ frac {1}} {\ frac {1} {1} {\ mathrm {\ eta}}}\ right) {\ mathrm {\ eta}}} _ {i} ^ {s}} ^ {s}}\ right) {\ mathit {fs}}} +\ left (-\ frac {1} {1} {\ frac {1} {\ mathrm {\ eta}}}\ right) {\ mathrm {\ eta}}}\ sigma}} ^ {s}\ end {array}\end{split}\]

In order to simplify the calculations, the following intermediate variables are defined:

(2.11)\[ {u} _ {\ mathit {rr}}\ equiv\ frac {{k}} _ {r} ^ {s}} {{\ mathrm {\ eta}} _ {r} ^ {s}}} =\ frac {1} {1} {{\ mathrm {\ tau}}} _ {r} ^ {s}}\]

The system of equations () can then be put into the following matrix form:

(2.12)\[\begin{split} {\ dot {\ mathrm {\ epsilon}}}}} ^ {\ mathit {fs}}} =\ left\ {\ begin {array} {c} {\ dot {\ mathrm {\ epsilon}}}} _ {r}}} _ {r}}} _ {r}}} _ {r}} ^ {r} ^ {\ mathit {fs}} {\ mathit {fs}}}\\ {\ mathit {fs}}}\\ {\ dot {\ mathrm {\ epsilon}}}} _ {i}}} _ {\ epsilon}}} _ {\ epsilon}}} _ {\ epsilon}}} _ {\ epsilon}}} _ {\ epsilon}}} _ {\ epsilon} fs}}\ end {array}\ right\} =\ left [\ begin {array} {} {cc} - {u} _ {\ mathit {rr}} -4 {u} _ {\ mathit {ri}}} & 2 {u}} & 2 {u}} _ {\ mathit {ri}}} & - {\ mathit {ri}}} & - {u} _ {\ mathit {ri}}} & - {\ mathit {ri}}} & - {u} _ {\ mathit {ri}}} & - {\ mathit {ri}}} & - {u} _ {\ mathit {ri}}} & - {\ mathit {ri}}} & - {u} _ {\ mathit {ri}}} & - {\ mathit {ri}}} & {array}\ right]\ left\ {\ begin {array} {c} {c} {\ mathrm {\ epsilon}} _ {r} ^ {\ mathit {fs}}\\ {\ mathrm {\ epsilon}}} _ {\ epsilon}}} _ {i}} ^ {i} ^ {i} ^ {i} ^ {\ mathit {fs}} ^ {\ mathit {fs}}}\ end {array}\ right\}} +\ frac {{\ mathrm {\ epsilon}}} _ {\ epsilon}} _ {i}} ^ {i} ^ {i} ^ {\ mathit {fs}} ^ {\ mathit {fs}} ^ {s}} {{k} _ {r} ^ {s}}}\ left\ {\ begin {s}}}\ left\ {\ begin {array} {c} {u}} +2 {u} _ {\ mathit {ri}}}\\ mathit {ri}} _\\ - {u} _ {\ mathit {ri}} _ {\ mathit {ri}}} +2 {u} _ {\ mathit {rr}} _ {\ mathit {rr}} _ {\ mathit {rr}} _ {\ mathit {ri}} _ {\ mathit {ri}} _ {\ mathit {ri}} right\}\end{split}\]

That is to say:

(2.13)\[ {\ dot {\ mathrm {\ epsilon}}}}} ^ {\ mathit {fs}} =A\ times {\ mathrm {\ epsilon}} ^ {\ mathit {fs}}}} + {\ mathrm {\ sigma}}} ^ {s} B\]

Assume that matrix \(A\) is*diagonalizable* (this property will be checked later):

(2.14)\[ A=PD {P} ^ {-1}\]

Where \(D\) refers to the diagonal matrix of the eigenvalues of the matrix \(A\), \(P\) the matrix of the eigenvectors of the matrix \(A\), and \({P}^{-1}\) the inverse matrix of the matrix \(P\). By making the product by quantity \({P}^{-1}\) over and over again, the relationship () can be put in the form:

(2.15)\[ {\ dot {\ mathrm {\ epsilon}}}}} ^ {\ mathit {fs}},\ text {*}} =D\ times {\ mathrm {\ epsilon}}} ^ {\ mathit {fs}},\ text {fs}},\ text {*}},\ text {*}} + {\ mathrm {*}}} + {\ mathrm {\ sigma}} ^ {\ mathrm {\ epsilon}} ^ {\ mathit {fs}},\ text {*}} + {\ mathrm {\ sigma}}} ^ {\ text {\ epsilon}} ^ {\ mathit {fs}},\ text {*}}\]

Let \({\mathrm{\lambda }}_{1}\) and \({\mathrm{\lambda }}_{2}\) be the eigenvalues of the matrix \(A\). Quantities are defined:

(2.16)\[ {\ mathrm {\ epsilon}} ^ {\ mathit {fs},\ text {*}}\ equiv\ left\ {\ begin {array} {c} {\ mathrm {\ epsilon}}} _ {\ epsilon}} _ {\ epsilon}} _ {\ epsilon}} _ {\ epsilon}} _ {\ epsilon}} _ {\ epsilon}} _ {\ epsilon}} _ {\ epsilon}} _ {\ epsilon}} _ {\ epsilon}} _ {\ epsilon}} _ {\ epsilon}} _ {\ epsilon}} _ {\ epsilon}} _ {\ epsilon}} _ {\ epsilon}} _ {\ epsilon}}\ end {array}\ right\}\]

The relationship () is then written:

(2.17)\[\begin{split} \ {\ begin {array} {c} {\ dot {\ mathrm {\ epsilon}}}} _ {1} ^ {\ text {*}} (t) = {\ mathrm {\ lambda}}} _ {1} {1} {\ mathrm {\ epsilon}}} _ {1} ^ {\ text {*}}} (t) = {\ mathrm {\ lambda}}} _ {1} {\ mathrm {\ sigma}}} ^ {s} {b} _ {1} ^ {\ text {*}}}\\ {\ dot {\ mathrm {\ epsilon}}}} _ {2} ^ {\ text {*}} (t) = {\ mathrm {\ lambda}}} = {\ mathrm {\ lambda}}} _ {2} ^ {\ text {*}} (t) = {\ mathrm {\ lambda}}} _ {2} ^ {\ text {*}} (t) = {\ mathrm {\ lambda}}} _ {2} ^ {\ text {*}} (t) = {\ mathrm {\ lambda}}} _ {2} ^ {\ text {*}} (t) = {\ mathrm {\ lambda}}} {\ mathrm {\ sigma}} ^ {s} {s} {b} {b} _ {2} ^ {\ text {*}}}\ end {array}\end{split}\]

System whose solution is written as:

\[\]

: label: eq-20

{begin {array} {c} {mathrm {epsilon}}} _ {1} ^ {text {*}} (t) =-frac {{mathrm {sigma}}} ^ {sigma}} ^ {s}}} ^ {s}}} ^ {s}}} ^ {s}}} ^ {s}}} ^ {s}}} ^ {s}}} {s}}} {{s}}} {{s}}} {{s}}} {{s}}} {{mathrm {lambda}}} {{mathrm {lambda}}} {{mathrm {lambda}}} _ {1}} _ {1}} ^ {text {*}}} + mathrm {mu}} _ {1}mathrm {exp}left ({mathrm {lambda}}} _ {1} tright)\ {mathrm {epsilon}}} _ {epsilon}}} _ {2}}} _ {2}} ^ {2} ^ {2} ^ {2} ^ {2} ^ {2} ^ {2} ^ {text {*}}} (t) =-frac {{mathrm {sigma}}} ^ {epsilon}}} _ {epsilon}}} _ {epsilon}} _ {2}} ^ {2} ^ {2} ^ {2} ^ {2} ^ {2} ^ {2} ^ {text {*}}mathrm {lambda}} _ {2}} {b}} {b} _ {2}} ^ {text {*}}} + {mathrm {mu}} _ {2}mathrm {exp}\ left ({mathrm {exp}}left ({mathrm {lambda}}} _ {2} tright)end {exp}left ({mathrm {exp}}left ({mathrm {lambda}}} _ {2} tright)end {exp}left ({mathrm {exp}}left (

We can then return to the initial space, through the passage matrix; the reversible and irreversible creep deformations are linear combinations of \({\mathrm{\epsilon }}_{1}^{\text{*}}\) and \({\mathrm{\epsilon }}_{2}^{\text{*}}\). The eigenvalues of the matrix \(A\) are obtained by solving:

\[\]

: label: eq-21

mathit {det}left (A- {mathrm {lambda}} _ {i} Iright) =0

Or again:

:math:`left

begin{array}{cc}-{u}_{mathit{rr}}-4{u}_{mathit{ri}}-{mathrm{lambda }}_{i}& 2{u}_{mathit{ii}}\ 2{u}_{mathit{ri}}& -{u}_{mathit{ii}}-{mathrm{lambda }}_{i}end{array}right

=0iff {mathrm{lambda }}_{i}^{2}+left({u}_{mathit{rr}}+4{u}_{mathit{ri}}+{u}_{mathit{ii}}right){mathrm{lambda }}_{i}+{u}_{mathit{rr}}{u}_{mathit{ii}}=0`

Noting that \({u}_{\mathit{rr}}\), \({u}_{\mathit{ri}}\), and \({u}_{\mathit{ii}}\) are strictly positive, the discriminant is therefore always strictly positive. The eigenvalues are therefore real and distinct, the matrix \(A\) is therefore diagonalisable. Moreover, neither of the two eigenvalues is equal to zero (\({\mathrm{\lambda }}_{1}{\mathrm{\lambda }}_{2}={u}_{\mathit{rr}}{u}_{\mathit{ii}}\ne 0\)). The two eigenvalues are defined by:

(2.18)\[ \ {\ begin {array} {c} {\ mathrm {\ lambda}}} _ {1}} =\ frac {-\ left ({u} _ {\ mathit {rr}} +4 {u} _ {\ mathit {ri}}} _ {\ mathit {ri}}}} + {u}}} + {u}} _ {u} _ {\ mathit {ii}}\ right) -\ sqrt {\ mathrm {\ delta}}} {\ delta}} _ {\ mathit {ri}}} {ri}}} {2}}\ {\ mathrm {\ lambda}} _ {2} =\ frac {-\ left ({u} _ {\ mathit {rr}} +4 {u} _ {\ mathit {ri}}} + {u}} _ {\ mathit {ii}}}\ right) +\ sqrt {\ mathit {ii}}}\ right) +\ sqrt {\ mathrm {\ delta}}} {2}\ end {array}} {2}\ end {array}}\]

We can show that both eigenvalues are actually negative. Let’s show that the second eigenvalue is negative. Spherical creep deformation is therefore asymptotic, a hypothesis put forward in the spherical proper creep model [bib1]. Now let’s determine a base of the eigenvectors \(\left({X}_{1},{X}_{2}\right)\) associated with the eigenvalues \({\mathrm{\lambda }}_{1}\) and \({\mathrm{\lambda }}_{2}\). It is determined by solving equation \(\left(A-{\mathrm{\lambda }}_{i}I\right){X}_{i}=0\). A particular eigenvector base is written as:

(2.19)\[\begin{split} {X} _ {1} =\ left\ {\ begin {array} {c} {x} _ {1}\\ 1\ end {array}\ right\}\end{split}\]

After verifying that \(P\) can actually be reversed, we deduce the solution in physical space:

(2.20)\[\begin{split} \ {\ begin {array} {c} {\ mathrm {\ epsilon}}} _ {r}} ^ {\ mathit {fs}} (t) =\ left [- {\ mathrm {\ sigma}}} ^ {\ sigma}}} ^ {s}} ^ {s}} ^ {\ mathrm}}} {{\ mathrm}}} {{\ mathrm}}} {{\ mathrm}}} {{\ mathrm {\ sigma}}} ^ {\ mathrm {\ sigma}}} ^ {\ mathrm {\ sigma}}} ^ {\ mathrm {\ sigma}}} ^ {\ mathrm {\ sigma}}} ^ {\ mathrm}}} {{\ mathrm}} {\ mathrm\ lambda}} _ {1}} +\ frac {{b} _ {2} _ {2}} ^ {\ text {*}}} {{\ mathrm {\ lambda}}} _ {2}}\ right) + {x} _ {1} {1} {\ mathrm {\ mu}}} _ {1}\ mathrm {exp}\ left ({\ mathrm {\ lambda}}\ left ({\ mathrm {\ lambda}}} _ {1} t\ right) + {\ mathrm {\ mu}}} _ {2}\ mathrm {exp}\ left ({\ mathrm {\ lambda}} _ {2} t\ right)\ right)\ right]\\ {\ mathrm {\ mu}}} _ {\ mathrm {\ epsilon}}} _ {i} ^ {\ mathit {fs}} (t) =\ left [- {\ mathrm {\ mathrm {\ epsilon}}} _ {\ mathit {fs}} (t) =\ left [- {\ mathrm {\ mu}}} (right)\ right]\\ right]\\ {\ mathrm {\ mu}}} _ {\ mathrm {\ mu}}} _ thrm {\ sigma}} ^ {s}\ left (\ frac {{b}} _ {1} _ {1} ^ {\ text {*}}}} {{\ mathrm {\ lambda}} _ {1}}} + {x}}} + {x}}} + {x}}} + {x}}} + {x}} _ {2}\ frac {{b} _ {2}}\ frac {{b} _ {2} _ {2}} ^ {\ text {*}}}} {{\ mathrm {\ lambda}}} _ {2}}\ right) + {\ mathrm {\ mu }} _ {1}\ mathrm {exp}\ left ({\ mathrm {\ lambda}}} _ {1} t\ right) + {x} _ {2} {\ mathrm {\ mu}}} _ {2}\ mathrm {exp}\ left ({\ mathrm {\ lambda}}} _ {2} t\ right)\ right)\ right]\ end {array}\end{split}\]

With:

(2.21)\[\begin{split} \ {\ begin {array} {c} {b} _ {1} _ {1} ^ {1} ^ {\ text {*}}} =\ frac {1} {x} _ {2} -1}\ left [{x} _ {2} _ {2} _ {2}}\ left ({u}}\ left ({u}} _ {\ mathit {ri}} -1}\ left [{x}} -1}\ left [{x} _ {2} -1}\ left [{x} _ {2} -1}\ left [{x} _ {2} -1}\ left [{x} _ {2} _ {2} -1}\ left [{x} _ {2} _ {x} -1}\ left [{x} _ {x} -1}\ left [{x} _ {_ {\ mathit {ri}}\ right]\\ {b} _ {2}} _ {2}} ^ {\ text {*}} =\ frac {1} {x} _ {2} -1}}\ left [-\ {b} -1}}\ left [-\ left ({u}} _ {\ mathit {rr}}} =\ frac {1} {x}} _ {right) - {2} -1}}\ left [-\ left ({u}} _ {1} {u} _ {\ mathit {ri}}}\ right]\ end {array}\end{split}\]

Finally, \({\mathrm{\mu }}_{1}\) and \({\mathrm{\mu }}_{2}\) are defined by relationships:

\[\]

: label: eq-27

{begin {array} {c} {mathrm {mu}}} _ {mu}}} _ {1} =-frac {1} {1} {x} _ {2} -1right)mathrm {exp}right)mathrm {exp}}leftleft [left ({mathrm {lambda}}} _ {1}} + {mathrm {lambda}} + {mathrm {lambda}} _ {2}right) {t} _ {0}right]}left [frac {1} {1} {2} {k} _ {2}mathrm {exp}left ({mathrm {exp}left ({mathrm {lambda}}}} _ {lambda}}} _ {2} {t} _ {right) -frac {1} {{1} {{k}}leftleft ({mathrm {lambda}}}} _ {i}}mathrm {exp}left ({mathrm {lambda}}} _ {2} {t} _ {0}right)right]\ {mathrm {mu}} _ {2}} _ {2}} _ {2}}} _ {2}}} _ {2}}} _ {2}}} _ {2}}} _ {2}}} _ {2}}} _ {2}}} _ {2}} _ {2}} _ {2}} _ {2}} _ {2}} _ {2}} _ {2}} _ {2}} _ {2}} _ {2}} _ {2}} _ {2}} _ {2}} _ {2}} _ {2}} _ {2}} left [left ({mathrm {lambda}}} _ {1}} _ {1}} + {mathrm {lambda}}} _ {0}right]}left [-frac {1}} {1} {1} {1} {2} {k}} _ {r}}}mathrm {exp}right) {mathrm {lambda}}}left ({mathrm {lambda}}} _ {1}} {t} _ {0}right ) +frac {1} {{k} _ {i}} {i}} {x}} _ {1}mathrm {exp}left ({mathrm {lambda}}} _ {1} {t} {t} _ {t} _ {0}right)right]end {array}

2.1.3. Solving the equations that make up deviatoric creep

Deviatory deformations include a reversible part and an irreversible part (cf. [R7.01.06]):

(2.22)\[ \ underset {\ text {total deviatoric deformation}}} {\ underset {} {{\ mathrm {\ epsilon}}} ^ {\ mathit {fd}}}} =\ underset {\ text {text {text {adsorbed water contribution}}} {\ underset {}} {\ underset {}} {\ underset {}} {\ mathrm {\ epsilon}}} _ {r} ^ {\ mathit {fd}}}} =\ underset {\ text {text {{text {contribution to adsorbed water}}}} {\ underset {}}} +\ underset {\ text {free water contribution}}} {\ underset {} {\ mathrm {\ epsilon}} _ {i} ^ {\ mathit {fd}}}}\]

The*th* main component of total deviatoric deformation is governed by equations () and ():

(2.23)\[ {\ mathrm {\ eta}} _ {r} ^ {d} {d} {\ dot {\ mathrm {\ epsilon}}}} _ {r} ^ {d, j} + {k} _ {r} ^ {d} ^ {d} ^ {d} ^ {d, j} + {k} _ {r}} _ {\ mathrm {\ sigma}} ^ {d, d}} ^ {d, j}} ^ {d, j}} ^ {d, j}} ^ {d, j}} ^ {d, j}} ^ {d, j}} ^ {d, j}} ^ {d, j}}}\]

where \({k}_{r}^{d}\) refers to the stiffness associated with the capacity of adsorbed water to transmit charges (load bearing water) and \({\mathrm{\eta }}_{r}^{d}\) is the viscosity associated with the water adsorbed by the hydrate sheets.

\[\]

: label: eq-30

{mathrm {eta}} _ {i} ^ {d}} {dot {mathrm {epsilon}}}} _ {i} ^ {d, j} =h {mathrm {sigma}}} ^ {d, j} =h {mathrm {sigma}}} ^ {d, j}

where

refers to the viscosity of free water. The system of equations () and () is simpler to solve than that governing spherical behavior because it is decoupled. It is always assumed that the humidity remains equal to 1 during the entire load. Equation () corresponds to the Kelvin visco-elastic model whose response to a stress level is of the exponential type. As for equation (), the deformation response is linear with time. The total creep deformation is therefore written as the sum of the contribution of a Kelvin chain and the contribution of a shock absorber and series:

\[\]

: label: eq-31

{mathrm {epsilon}} ^ {d, j} (t) =left [frac {t} {{mathrm {eta}} _ {i} ^ {d}}} +frac {1} {d}}} +frac {1} {d}}frac {{k}} =left (-frac {{k}} _ {r} ^ {d}} {{mathrm {sigma}} {{mathrm {eta}}} _ {r} ^ {d}}} tright)right] {mathrm {sigma}}}} ^ {d, j} H (t)

2.1.4. Analytical Solution Summary

For uniaxial loading, the analytical solutions of the two deformation components are known. The contribution of the deviatory part is written as:

(2.24)\[ {\ mathrm {\ epsilon}}} ^ {\ mathit {fd}} (t) =\ frac {2} {3} {\ mathrm {\ sigma}} _ {0}\ left [\ frac {t}} {\ frac {t}} {{t}} {{t}} {{t}} {t} {t} {t} {t} {t} {{t}} {{t}} {{t}} {{t}} {{t} {{t}} {{t}} {{t} {{t}} {{t}} {{t}} {{t} {{t}} {{t}} {{t} {{t}} {{t}} {{t} {{t}} {{t}}}\ left (1-\ mathrm {exp}}\ left (-\ frac {{k} _ {r} ^ {d}}} {{\ mathrm {\ eta}}} _ {r} {r} ^ {d}}\ left (-\ frac {{k}} _ {r} {\ mathrm {\ eta}}}} _ {r} ^ {d}}} _ {r} ^ {d}} _ {r} ^ {d}}} t\ right)\ right]\]

As for the contribution of the spherical part, the solution is defined over two intervals:

(2.25)\[ {\ mathrm {\ epsilon}}} ^ {\ mathit {fs}}} =\ {\ begin {array} {c}\ frac {{\ mathrm {\ sigma}} _ {0}} {3 {k}} {3 {k}} _ {k} _ {k} _ {k} _ {k} _ {k} _ {r} _ {r}}}}\ left [1-\ mathrm {exp}\ left (-\ frac {{k}}} {3 {k}} {k}} {3 {k}} _ {k} _ {k} _ {r}} _ {r} {s}}}\ left [1-\ mathrm {exp}\ left (-\ frac {{k}}} {3}} {s}} {{\ mathrm {\ eta}}} _ {r} ^ {s}} _ {r} ^ {s}} t\ right)\ right]\ text {for} t\ le\ frac {{\ mathrm {\ eta}}} _ {r} ^ {eta}}} _ {\ mathrm {\ eta}}} _ {\ mathrm {\ eta}}} _ {r} ^ {s}} _ {\ mathrm {\ eta}}} _ {r} ^ {s}} _ {\ mathrm {\ eta}}} _ {r} ^ {s}} _ {\ mathrm {\ eta}}} _ {r} ^ {s}}} _ {\ mathrm {\ eta}}} _ {r} ^ {s}} sigma}} _ {0}} {3}\ left [\ left (\ frac {1} {{k} _ {r} ^ {s}}} +\ frac {1} {{k} _ {i} ^ {s}} {s}}\ right)\ left) + {\ mathrm {\ mu}}}\ right)\ mathrm {\ mu}}} _ {1}\ right)\ mathrm {exu}} p}\ left ({\ mathrm {\ lambda}}} _ {1} t\ right) + {\ mathrm {\ mu}}} _ {2}\ left (1+ {x} _ {2}\ right)\ mathrm {exp}}\ left ({\ mathrm {\ lambda}}} _ {2} t\ right)\ right]\ text {for}\ right)\ text {for} t>\ right frac {{\ mathrm {\ eta}} _ {r} ^ {s}} {{s}} {{k}} _ {r} ^ {s}}\ mathrm {ln} 2\ end {array}\]

Axial deformation is a linear function of the two previous contributions:

(2.26)\[ {\ mathrm {\ epsilon}} _ {\ mathit {zz}}} = {\ mathrm {\ epsilon}}} ^ {\ mathit {fs}}\ left (\ frac {{\ mathrm {\ mathrm {\ sigma}}}} _ {\ sigma}}} _ {0}}} _ {0}}} {0}} {3}\ right) + {\ mathrm {\ epsilon}}} ^ {\ mathit {fd}}\ left (\ frac {2 {\ mathrm {\ sigma}}} _ {0}} {3}\ right)\]

2.2. Solutions obtained for model BETON_BURGER

The analytical solution was not developed for this uniaxial creep load. The reference solution is obtained numerically using a python script (accessible under the astest directory: SSNV163D .44). The integration scheme used is explicit and sensitive to the temporal discretization used.

2.3. Reference quantities and results

The test is homogeneous. The deformation is tested at any node.

2.4. Uncertainty about the solution

Exact analytical result for BETON_UMLV.

Result depending on the time discretization used for BETON_BURGER.

2.5. Bibliographical references

1. BENBOUDJEMA, F.: Modeling delayed deformations of concrete under biaxial stresses. Application to reactor buildings of nuclear power plants, Thesis of D.E.A. Advanced Materials — Structural and Envelope Engineering, 38 pp. (+ appendices) (1999).

2. BENBOUDJEMA, F., MEFTAH, F., HEINFLING, G., G., LE PAPE, Y.: Numerical and analytical study of the spherical part of the proper creep model UMLV for concrete, technical note HT-25/02/040/A, 56 p (2002).

3. LE PAPE, Y.: UMLV behavioral relationship for the clean creep of concrete, code_aster [R7.01.06] Reference Documentation, 16 p (2002).

4. FOUCAULT, A.: BETON_BURGER behavioral relationship for the clean creep of concrete, code_aster [R7.01.35] (2011) Reference Documentation.