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--- doc:user:elements:volumes:iso_hypo_materials [2015/07/14 14:40] – [KelvinVoigtViscoElastHypoMaterial] papeleux
+++ doc:user:elements:volumes:iso_hypo_materials [2022/07/14 11:04] – papeleux
@@ Line 17: / Line 17: @@
 | Young's modulus                                    |  ''ELASTIC_MODULUS''  |              |
 | Poisson Ratio                                      |   ''POISSON_RATIO''   |              |
+| Material Stiffness (0 : Ana - 1 : Num)             | ''MATERIALSTIFFMETHOD''  |      -       |
+|     (only if element Stiffness == STIFF_ANALYTIC)  |                       |       -      |
 | Objectivity Method \\ (Jaumann = 0, GreenNaghdi = 1) |    ''OBJECTIVITY''    |      -       |
 | Orthotropic axis                                   |    ''ORTHO_AX1_X''    |      -       |
@@ Line 64: / Line 66: @@
 \begin{cases}
 p^{1}  = p^{0} + 3K {\Delta\epsilon}_{ii} \\
-s^{1}_{ij}  = s^{0}_{ij} + 2G {\Delta\epsilon}_{ij} + \eta \frac{{\Delta\epsilon}_{ij}}{\Delta t}
+s^{1}_{ij}  = s^{0}_{ij} + 2G {\Delta\hat{\epsilon}}_{ij} + \eta \frac{{\Delta\hat{\epsilon}}_{ij}}{\Delta t}
 \end{cases}
 $$
@@ Line 80: / Line 82: @@
 | Viscosity coefficient                             |   ''VISCO_ETA''       |    TO/TM     |
+===== GeneralizedMaxwellViscoElastHypoMaterial=====
+=== Description ===
+The GeneralizedMaxwell viscoelastic model result in adding of up to now, maximum 2 visco-elastic Maxwell branches to the elastic material. The similar spring/damper model is shown at the figure below
+{{ :doc:user:elements:volumes:generalizedmaxwell.png?400 |}}
+Defining Maxwell viscous parameters from materials data (for each Maxwell branch):
+$$
+\begin{cases}
+\Gamma_{i} &= \frac{\mu_{i}}  {2G}\\
+\tau_{i}   &= \frac{\eta_{i}} {\mu_{i}}
+\end{cases}
+$$
+The stresses in each Maxwell branch consist in 2 effects :
+  - the relaxation of previous time step stresses in this Maxwell branch
+  - the stress modification due to strain increment ${\Delta\epsilon}_{ij}$
+$$
+\begin{cases}
+p^{1}      &= p^{0} + 3K {\Delta\epsilon}_{ii} \\
+\underline{s}^{1}  &= \underline{s}_{E}^{1} + \underline{s}^{1}_{M1} + \underline{s}^{1}_{M2}
+\end{cases}
+$$
+Stresses in each branch are computed using :
+$$
+\begin{cases}
+\underline{s}_{E}^{1}  &= \underline{s}_{E}^{0} + 2G {\Delta\hat{\underline{\epsilon}}} \\
+\underline{s}^{1}_{M1} &= e^{(\frac{-\Delta t}{\tau_{1}})} \underline{s}^{0}_{M1} + \Gamma_{1} (1-e^{\frac{-\Delta t}{\tau_{1}}}) \frac{\tau_{1}}{\Delta t} 2G {\Delta\hat{\underline{\epsilon}}} \\
+\underline{s}^{1}_{M2} &= e^{(\frac{-\Delta t}{\tau_{2}})} \underline{s}^{0}_{M2} + \Gamma_{2} (1-e^{\frac{-\Delta t}{\tau_{2}}}) \frac{\tau_{2}}{\Delta t} 2G {\Delta\hat{\underline{\epsilon}}} \\
+\end{cases}
+$$
+where $K$ and $G$ are compressibility and shear modulus, ${\Delta\underline{\epsilon}}$ is the stain increment during the time step  $\Delta t$.
+these equations shown clearly the necessity to track history of the total stresses but also to each Maxwell branch stresses $\underline{s}^{1}_{M1}$ and $\underline{s}^{1}_{M2}$.
+=== Parameters ===
+^   Name                                             ^     Metafor Code     ^  Dependency  ^
+| Density                                           |   ''MASS_DENSITY''    |      -       |
+| Young's modulus                                   |  ''ELASTIC_MODULUS''  |    TO/TM     |
+| Poisson Ratio                                     |   ''POISSON_RATIO''   |    TO/TM     |
+| Number of Maxwell branches 1 (default) or 2       |  ''NBMAXWELLBRANCH''  |      -       |
+| Maxwel 1 Stiffness                                |   ''VISCO_MU1''       |      -       |
+| Maxwel 1 Viscosity                                |   ''VISCO_ETA1''      |      -       |
+| Maxwel 2 Stiffness                                |   ''VISCO_MU2''       |      -       |
+| Maxwel 2 Viscosity                                |   ''VISCO_ETA2''      |      -       |
+model implemented based on {{:doc:user:references:materials:1997_formulation_and_implementation_of_three-dimensional_viscoelasticity_at_small_and_finite_strains_kaliske_rothert.pdf|Kaliske M,
+Rothert H. Formulation and implementation of three-dimensional viscoelasticity at small and finite strains. Computational Mechanics 1997;19:228-239.}}
 ===== EvpIsoHHypoMaterial =====
@@ Line 294: / Line 352: @@
 $$
-f=\dfrac{\overline{\sigma}}{1-D}-\sigma_{yield}=0
+f=\dfrac{\overline{\sigma}}{1-h\cdot D}-\sigma_{yield}=0
 $$
-where $ \overline{\sigma} $ is the equivalent stress computed as a function of [[doc:user:elements:volumes:plasticity_criterion#vonmisesplasticcriterion|Von Mises plasticy criterion]], $ \sigma_{yield} $ is the yield stress, $ D $ is the damage variable updated as a function of the [[doc:user:elements:volumes:continuousdamage|damage evolution law]].
+where $ \overline{\sigma} $ is the equivalent stress computed as a function of [[doc:user:elements:volumes:plasticity_criterion#vonmisesplasticcriterion|Von Mises plasticy criterion]], $ \sigma_{yield} $ is the yield stress, $ D $ is the damage variable updated as a function of the [[doc:user:elements:volumes:continuousdamage|damage evolution law]]. Moreover, $h$ is the Micro-Crack Closure Effect parameter that makes the distinction of the weakening effect of damage under compressive and tensile stress states, which is defined as:
+$$
+ h = \left\{
+ \begin{array}{ll}
+  \text{DAMAGE_MCCE} &\mbox{, if } \dfrac{p}{J_2}< 0.0\\
+.0 &\mbox{, if } \dfrac{p}{J_2}\geq 0.0\\
+ \end{array}
+ \right.
+$$
 The evolution law coupled with plasticity can be integrated three ways depending on the parameter ''TYPE_INTEG'':
@@ Line 315: / Line 381: @@
 | Initial damage                         |     ''DAMAGE_INIT''     |    -     |
 | Integration method                         |     ''TYPE_INTEG''      |    -     |
+| Micro-Crack Closure Effect parameter (=1.0 by default)   |    ''DAMAGE_MCCE''     | |
 ===== ContinuousAnisoDamageEvpIsoHHypoMaterial =====