4. Digital resolution#

4.1. Evaluation of the internal variable Y#

The calculation for \(Y\) is very simple and follows an explicit pattern. The steps are as follows:

  • Calculation of elastic and thermal deformations

  • Calculation of the main elastic stresses and evaluation of \(\gamma\).

  • Calculation of the equivalent deformation (and).

Calculation of variables :math:`r`, :math:`A`*and \(B\)

  • Calculation of the internal variable \(Y\).

    • If \(Y\mathrm{\le }{Y}^{\mathrm{-}}\) then \({Y}^{+}\mathrm{=}{Y}^{\mathrm{-}}\).

If \(Y>{Y}^{\mathrm{-}}\) then \({Y}^{+}\mathrm{=}Y\) .

Note: Currently the variable stored during the calculations is \({\varepsilon }_{\text{eq}}\) in order not to change existing couplings with UMLV. A condition on the strictly increasing evolution of the damage allows this simplification in the case where \(\gamma\) varies.

4.2. Assessment of damage#

The damage is calculated in all cases using the equation.

\[\]

: label: EQ-None

D=1-frac {left (1-Aright) {Y} _ {0}} {0}} {Y} -Atext {exp}left (-Bleft (Y- {Y} _ {Y} _ {0}right)right)

It is important to specify that we require \(D\) to be between 0 and 1 because it is possible to have values outside this framework depending on the choice of material parameters as for the original model.

4.3. Stress calculation#

After evaluating \(D\), we simply calculate:

\[\]

: label: EQ-None

sigmamathrm {=} (1mathrm {-} D)mathrm {A} {varepsilon} ^ {mathrm {e}}

4.4. Calculation of the tangent matrix#

One of the disadvantages of the Mazars model is the absence of a tangent matrix. It is not possible to calculate this matrix because of the use of the MacCauley operator in calculating the equivalent deformation, of \(\gamma\) and \(r\). However, it is possible to use an approximation during radial loads.

We are looking for the \(\mathrm{M}\) tensor such as \(\dot{\sigma }\mathrm{=}\mathrm{M}\dot{\varepsilon }\) knowing that \(\sigma \mathrm{=}(1\mathrm{-}D)\mathrm{A}\varepsilon\). The matrix is therefore the sum of two terms, one with constant damage, the other due to the evolution of the damage:

\[\]

: label: EQ-None

dot {sigma}mathrm {=} (1mathrm {-} D)mathrm {-} D)mathrm {-}mathrm {-}mathrm {A}mathrm {A}varepsilondot {D}

The first term is easy, it’s the Hooke operator, multiplied by the factor \(1\mathrm{-}D\).

The second requires the evaluation of damage increment \(\dot{D}\).

If a radial loading is imposed, the variables \(\gamma\), \(r\),, \(A\), and \(B\) are constant. By posing:

\[\]

: label: EQ-None

dot {D}mathrm {=}frac {mathrm {partial} D} {mathrm {partial} Y}frac {mathrm {partial} Y} {mathrm {partial}} {mathrm {partial}} (mathrm {partial}} (mathrm {partial}} (gamma {varepsilon} _ {text {eq}}})} {mathrm {partial}varepsilon}dot {varepsilon}dot {varepsilon}

With

\[\]

: label: EQ-None

frac {mathrm {partial} Y} {mathrm {partial}} {mathrm {partial}} (gamma {varepsilon}})}frac {mathrm {partial} (gamma {partial}} (gamma {varepsilon}} _ {text {varepsilon}})} {mathrm {partial}partial} (gamma {varepsilon} _ {text {eq}})} {mathrm {partial}partial}partial} (gamma {varepsilon})} {mathrm {partial}partial}partial} (gamma {varepsilon}}mathrm {=}frac {gamma {gamma {mathrm {langle}varepsilonmathrm {rangle}} _ {+}} {{epsilon}} _ {epsilon} _ {text {eq}}}

Under this radial load condition, the deformation increment is written as:

\[\]

: label: EQ-None

dot {D}mathrm {=}left [frac {(1mathrm {-} A) {Y} _ {0}} {{Y} ^ {2}} +text {AB}\ text {AB}\ text {exp} (B (Ymathrm {-} -} {0}))right]frac {gamma {mathrm {mathrm {mathrm {langle}varepsilonmathrm {rangle}}} _ {+}} {{varepsilon} _ {text {eq}}}}dot {varepsilon}

Notes:

1.

Given the simplifications made, in the general case the tangent matrix is not consistent. Also, it may happen that updating the tangent matrix during Newton iterations does not help convergence. In this case, it is sufficient to use only the secant matrix by enforcing STAT_NON_LINE (NEWTON = _F (REAC_ITER =0)) .

2.

In the general case, the tangent matrix is non-symmetric. It is**possible to do this with the keyword ** SOLVEUR =_F (SYME = “OUI”) * of STAT_NON_LINE * .

3 .

The analytic expression for the tangent matrix is only valid for radial loads ( \(\mathit{dr}\mathrm{=}d\gamma \mathrm{=}0\) ). In other cases, the quadratic convergence of the method is no longer guaranteed.

4.5. Stored internal variables#

In the following table, we indicate the internal variables stored at each Gauss point for the MAZARS model:

Internal variable

Physical sense

\(\mathit{V1}\)

\(D\): damage variable

\(\mathit{V2}\)

damage indicator (0 if elastic, 1 if damaged, i.e. as soon as \(D\) is no longer zero)

\(\mathit{V3}\)

\(\mathit{Tmax}\): maximum temperature \(\theta\) reached at the gauss point

\(\mathit{V4}\)

\({\varepsilon }_{\text{eq}}\mathrm{=}\sqrt{{\mathrm{\langle }\varepsilon \mathrm{\rangle }}_{+}\mathrm{:}{\mathrm{\langle }\varepsilon \mathrm{\rangle }}_{+}}\) the equivalent deformation

Table 4.5-1 : Stored internal variables.