3. Digital integration

3.1. Discretized equations

The time discretization of behavioral equations is based on a schema of order 1. We note \({Q}_{n}\) the value of a quantity \(Q\) at the beginning of the time step, \(\mathrm{\Delta }Q\) its increment during the time step and (simply) \(Q\) its value at the end of the time step. The mechanical state at the start of time step \(({\epsilon }_{n},{{\sigma }^{v}}_{n})\) is assumed to be known as well as the deformation increment \(\mathrm{\Delta }\epsilon\) (and therefore also the deformation \(\epsilon\)). It is then a question of calculating the constraint \({\sigma }^{v}\) at the end of the time step.

The discretization of the stress-strain relationship () is written as follows:

\[\]

: label: eq-5

{dot {mathrm {sigma}}}}} ^ {v}} ^ {v} +frac {1} {mathrm {tau}} {sigma}} ^ {v} =kfrac {frac {mathrm {delta} +frac {delta} +frac {delta}}text {;} {mathrm {sigma}} ^ {v} =kfrac {mathrm {sigma}} =kfrac {sigma}} =kfrac {mathrm {sigma}} =kfrac {sigma}} =kfrac {mathrm {sigma}} =kfrac {sigma}} =kfrac {mathrm {{sigma}} ^ {v} ({t} _ {n}) = {{mathrm {sigma}} ^ {v}}} _ {n}

The second member is then constant, so that the solution to this differential equation is written as:

(3.1)\[ {\ mathrm {\ sigma}} ^ {v}} = {A} = {A} _ {\ mathrm {\ Delta} t} {{\ mathrm {\ sigma}} ^ {v}} _ {n} + {B}} _ {B} _ {B} _ _ {B} _ _ {\ B} _ {_} _ {\ mathrm {\ Delta} t} =\ mathrm {exp}\ left (-\ frac {\ mathrm {\ delta} t} {\ mathrm {\ tau}}\ right)\ text {;} {B}} _ {\ mathrm {\ exp}\ left (-\ left) (-\ frac {\ delta} t} =\ frac {\ mathrm {\ tau}}\ right)\ text {;} {;} {B}} _ {\ mathrm {\ Delta} t} _ {\ mathrm {\ Delta} t}\ left [1-\ mathrm {exp}\ left (-\ frac {\ mathrm {\ Delta} t} {\ mathrm {\ tau}}}\ right)\ right)\ right]\]

The calculation of the viscous stress is therefore explicit.

3.2. Tangent matrix

The derivative of the expression () immediately provides the tangent matrix:

(3.2)\[ \ frac {\ mathrm {\ sigma}}} ^ {v}}} {d\ mathrm {\ epsilon}} = {B} _ {\ mathrm {\ Delta} t} {I} _ {4}\]

where \({I}_{4}\) refers to the fourth-order identity tensor.

In the perspective of a prediction by tangent matrix, the term of order 0 of the limited development of the discretized law of behavior () does not correspond to the viscous stress at the start of the time step but to:

(3.3)\[ {\ mathrm {\ sigma}} ^ {0} = {A} _ {\ mathrm {\ delta} t} {{\ mathrm {\ sigma}}} ^ {v}} = {v}}} _ {n}\]

This is the constraint to be provided to option RIGI_MECA_TANG.

3.3. Calculation of energies

The stored energy volume () is an explicit function of viscous stresses and poses no difficulty. The energy volume dissipated by viscosity () introduces an integral that will be evaluated by a mid-point method (which is compatible with order 1 of the integration diagram of the law of behavior):

(3.4)\[ \ text {VISCDISS}\ mathrm {:} =\ text {VISCDISS} =\ text {} +\ frac {\ mathrm {\ delta} t} {k\ mathrm {\ tau}} {\ left [\ frac {{{\ mathrm {{\ mathrm {\ sigma}}} {\ mathrm {\ tau}}} {\ left [\ frac {{{\ mathrm {{\ mathrm {\ sigma}} {\ mathrm {\ tau}}} {\ left [\ frac {{{\ mathrm {\ sigma}} {{\ mathrm {\ tau}}} {\ left [\ frac {{{\ mathrm {\ sigma}} {\ mathrm {\ tau}} {\ left [\ frac {^ {2}\]