5. Digital integration

In this paragraph, the numerical integration of the consideration of distributed reinforcements is presented. The model was integrated according to an implicit pattern.

5.1. Digital integration diagram

The implicit integration scheme consists of a visco-elastic prediction phase that takes creep into account. It is followed by a plastic correction phase in the space of effective constraints if at least one of the criteria is met.

The pseudo-vector notation (symmetric tensor) is adopted.

5.1.1. Viscoelastic prediction phase

The following three equations relate the elastic deformation and creep increments (Maxwell and Kelvin) to the total deformation. The schema is semi-implicit and \(\theta\) is worth 1/2:

\[\]

: label: eq-28

left{begin {array} {c}mathrm {Delta} {Delta} {epsilon} ^ {r,mathit {tot}} =mathrm {Delta} {epsilon} {epsilon}} ^ {epsilon} {epsilon} {epsilon}} =mathrm {Delta} {epsilon} ^ {r, K} {epsilon} ^ {epsilon} {epsilon} ^ {epsilon} ^ {epsilon} {epsilon} ^ {epsilon} +mathrm {Delta} {Delta} {epsilon} ^ {epsilon} +mathrm {Delta} {Delta} {epsilon} ^ {r, M}\mathrm {delta} {delta} {epsilon} ^ {r, K}left (frac {mathrm {Delta} t}} {{tau}} {{tau}} ^ {tau} ^ {tau} ^ {tau} ^ {tau} ^ {tau} ^ {tau} ^ {tau} ^ {r, K}} ^ {tau} ^ {tau} ^ {tau} ^ {r, K}} ^ {tau} ^ {tau} ^ {tau} ^ {r, K}} ^ {tau} ^ {tau} ^ {tau} ^ {r, K}} ^ {tau} ^ {tau} ^ {tau} ^ {Psi}} ^ {K}}left ({epsilon}} ^ {r,mathit {el}} +thetamathrm {Delta} {epsilon} ^ {r,mathit {el}}}right)\ mathit {el}}}right)\mathrm {el}}right)\mathrm {Delta} {epsilon} ^ {r, M} =frac {mathrm {el}}}right)\ mathrm {Delta} {epsilon} =frac {mathrm {el}}}right)\ mathrm {Delta} {epsilon}} =frac {mathrm {el}}}} t} {{tau} ^ {r, M} {mathit {Cc}} {mathit {Cc}}} ^ {tau} ^ {r,mathit {el}}} +thetamathrm {Delta} {Epsilon} ^ {r,mathit {el}}}right)end {array}right}

By synthesizing:

(5.1)\[ \ phantom {1}\ iff\ overline {\ overline {L}}\ cdot\ mathrm {\ Delta}\ overline {X} =\ overline {Y} =\ overline {Y}\]

With \(\overline{\overline{L}}\) established as the main base of \(d\text{'}\epsilon\) and \(\overline{X}=\left(\begin{array}{c}d{\epsilon }^{r,\mathit{el}}\\ d{\epsilon }^{r,M}\\ d{\epsilon }^{r,K}\end{array}\right)\)

Thus, we have:

\[\]

: label: eq-30

phantom {1}iffoverline {X} = {overline {overline {L}}} ^ {-1}cdotoverline {X} = {Y}

\[\]

: label: eq-31

{overline {X}} ^ {t} = {overline {X}}} ^ {-t} +mathrm {Delta} {overline {X}}} ^ {t}

\[\]

: label: eq-32

mathrm {Delta} {overline {X}}} ^ {t}}tomathrm {Delta} {epsilon} ^ {r,mathit {el}}

The constraint increment is written as:

(5.2)\[ \ mathrm {\ Delta} {\ sigma} {\ sigma} ^ {r} ^ {r} = {E} ^ {r}\ cdot\ mathrm {\ Delta} {\ epsilon}} ^ {epsilon} ^ {r,\ r,\ mathit {el}}\]

So:

(5.3)\[ {\ sigma} ^ {r} = {\ sigma} = {\ sigma} _ {0} _ {0} ^ {r} +\ mathrm {\ delta} {\ sigma} {\ sigma} ^ {r}\]

5.1.2. Correction phase

After the viscoelastic prediction phase, an evaluation of the criteria is carried out:

(5.4)\[ {f} ^ {r} =\ mathrm {\ Delta} {\ delta} {\ sigma} ^ {r} - {f} _ {y} ^ {r}\]

With:

(5.5)\[ {\ mathrm {\ Delta}\ sigma}} ^ {r} = {\ sigma} ^ {r} - {E} ^ {r}\ cdot\ mathrm {\ Delta} {\ Delta} {\ epsilon} {\ epsilon}} ^ {r} ^ {r,\ mathit {pl}}\]

If the criterion is activated, a flow is produced by considering the correction for the associated creep:

(5.6)\[ {f} _ {y} ^ {r} -\ mathrm {\ Delta} {\ delta} {\ sigma} ^ {r} =- {f} ^ {r}\]

With an associated flow law:

(5.7)\[ \ mathrm {\ Delta} {\ epsilon} {\ epsilon} ^ {r} ^ {r}} =\ frac {{f} ^ {r}} {{E} ^ {r} ^ {r} {r} {r} {r} {r}}\]

Stress variations cause a viscoelastic correction, which makes it necessary to couple the viscoelastic firing equations with those of radial return, the following system is then obtained and solved:

(5.8)\[\begin{split} [\ overline {Z}]\ left (\ begin {array} {c} {c} d {\ epsilon} _ {e}\\ d {\ epsilon} _ {M}\\ d {\ epsilon} _ {K}\\ k}\\ d {\ epsilon} {c}\\ d {\ epsilon} {c}\\ d {\ epsilon} _ {K}\\ K}\\ d {\ epsilon} _ {K}\\ k}\\ d {\ epsilon} _ {K}\\ k}\\ d {\ epsilon} _ {K}\\ k}\\ d {\ epsilon} _ {K}\\ k}\\ d {\ epsilon} _ {K}\\ k}\\} 0\\ 0\\ 0\\ 0\\ - {f} ^ {r}\ end {array}\ right)\end{split}\]

With \([\overline{Z}]\) the matrix containing all the coefficients deduced from the conditions for cancelling criteria and viscoelastic behavior.

Increments are used to get the allowable values of state variables at the end of the step. These can then be used to calculate damage in accordance with the resolution algorithm.