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# Continuous isotropic damage

The ContinousDamage class manages all continuous damage evolution laws. When a new law is defined, the evolution of the damage variable $\delta D$ must be defined, and so must be its derivatives with respect to pressure, plastic strain and damage.

Laws implemented in Metafor

## LemaitreChabocheContinuousDamage

#### Description

Lemaitre & Chaboche damage model [1,2]. $$\dot D = \left(\dfrac{\bar \sigma^2 R_\nu}{2ES\left(1-D\right)^2}\right)^s \dot \varepsilon^{pl} \mbox{, if } \varepsilon^{pl} > \varepsilon^{pl}_D \mbox{, and } \eta > \eta_D$$

where the triaxiliaty function is defined as:

$$R_\nu = \dfrac{2}{3}\left(1+\nu\right) + 3\left(1-2\nu\right) \left(\dfrac{p}{\bar\sigma}\right)^2$$

where $p$ is the pressure, $\bar \sigma$ is Von Mises stress and $\eta$ is the stress triaxiality ratio.

#### Parameters

Name Metafor Code Dependency
Young Modulus $E$ LEMAITRE_YOUNG TO/TM
Poisson Ratio $\nu$ LEMAITRE_NU TO/TM
Exponent $s$ LEMAITRE_SMALL_S TO/TM
Coefficient $S$ LEMAITRE_BIG_S TO/TM
Plastic strain threshold $\varepsilon^{pl}_D$ LEMAITRE_EPL_THRESHOLD TO/TM
Triaxiality threshold $\eta_D$ LEMAITRE_TRIAX_THRESHOLD TO/TM

## BoneRemodContinousDamage

This law is designed for bone remodeling (extracted from Doblaré's law, which he uses only in elasticity). Damage evolution depends mostly on damage, surface available for remodeling and a “remodelling rate” function, which depends on stress state.

#### Description

$$\dot d =f(d, \rho_0)kS_v(d)\dot r$$

where

$S_v(d)$ is the surface per unit volume available for remodeling (polynomial of degree 5 in $d$)

and where

\begin{align*} \dot r &= \ cf_1(d, \rho_0)g_f&\text{ if }g_f>0 \\ \dot r &= -cf_1(d, \rho_0)g_r&\text{ if }g_r>0 \end{align*} with $$\begin{eqnarray*} g_f &=& N^{1/4}u(\sigma)-(1+\omega)\psi \\ g_r &=& \dfrac{1}{N^{1/4}u(\sigma)}-\dfrac{1}{(1-\omega)\psi} \end{eqnarray*}$$ $f, f_1$ are functions in the damage variable, $u$ is a measure of the elastic strain energy.

#### Parameters

Name Metafor Code
Coefficient $N$ BONE_REMOD_N
Percentage of available surface $k$ BONE_REMOD_K
Reference elastic strain energy $\psi$ BONE_REMOD_PSI
Half width of the dead zone $\omega$ BONE_REMOD_OMEGA
Remodeling speed $c$ BONE_REMOD_C
Density of undamaged material$\rho_0 [kg/m^3]$ BONE_REMOD_MASS_DENSITY

### AlvBoneRemodContinousDamage

This law is defined for the remodeling of the alveolar bone. Damage evolution also depends on pressure.

#### Description

$$\dot d =f(d, \rho_0)kS_v(d)\dot r$$

where

$S_v(d)$ is the surface per unit volume available for remodeling (polynomial of degree 5 in $d$)

\begin{align*} \dot r &= cf_1(d, \rho_0)g_f &\text{ if }g_f>0 \text{ and } p>0 \\ \dot r &= -cf_1(d, \rho_0)g_f &\text{ if }g_f>0 \text{ and } p<0 \\ \dot r &= -cf_1(d, \rho_0)g_r &\text{ if }g_r>0 \end{align*} with $$\begin{eqnarray*} g_f &=& N^{1/4}u(\sigma)-\psi \\ g_r &=& \dfrac{1}{N^{1/4}u(\sigma)}-\dfrac{1}{\psi} \end{eqnarray*}$$ $f, f_1$ are functions in the damage variable, $u$ is a measure of the elastic strain energy.

#### Parameters

Name Metafor Code
Coefficient $N$ BONE_REMOD_N
Percentage of available surface $k$ BONE_REMOD_K
Reference elastic strain energy $\psi$ BONE_REMOD_PSI
Remodeling speed $c$ BONE_REMOD_C
Density of undamaged material $\rho_0 [\mbox{kg}/\mbox{m}^3]$ BONE_REMOD_MASS_DENSITY

### AlvBoneRemodContinousDamage2constant

Same law than the previous one, except that remodeling constants are different in formation and resorption.

#### Description

\begin{align*} \dot r &= c_ff_1(d, \rho_0)g_f\;\;&\text{ if }g_f>0\;\;\text{ and } p>0 \\ \dot r &= -c_rf_1(d, \rho_0)g_f\;\;&\text{ if }g_f>0\;\;\text{ and } p<0 \\ \dot r &= -c_rf_1(d, \rho_0)g_r\;\;&\text{ if }g_r>0 \end{align*} with <note> some definitions are lacking </note> ... === Parameters === ^ Name ^ Metafor Code ^ | Coefficient $N$ | BONE_REMOD_N |

 Percentage of available surface $k$ BONE_REMOD_K Reference elastic strain energy $\psi$ BONE_REMOD_PSI Remodeling speed $c_f$ BONE_REMOD_CF Remodeling speed $c_r$ BONE_REMOD_CR Density of undamaged material $\rho_0 [\mbox{kg}/\mbox{m}^3]$ BONE_REMOD_MASS_DENSITY

## LangsethContinousDamage

#### Description

$$\dot D = D_C\dfrac{\dot \varepsilon^{pl}}{\varepsilon^{pl}_f-\varepsilon^{pl}_D} \mbox{ if } \varepsilon^{pl} > \varepsilon^{pl}_D$$

where the plastic strain at failure is defined as:

$$\varepsilon^{pl}_f = \left(D_1 + D_2 \exp\left(D_3\dfrac{p}{\bar\sigma}\right)\right) \left(1+\ln\dfrac{\dot \varepsilon^{pl}}{\dot \varepsilon^{pl}_0}\right)^{D_4} \left(1-D_5\dfrac{T-T_{room}}{T_{melt}-T_{room}}\right)$$

where $p$ is the pressure and $\bar \sigma$ the Von Mises stress.

#### Parameters

Name Metafor Code Dependency
$D_1$ LANGSETH_D1 TO/TM
$D_2$ LANGSETH_D2 TO/TM
$D_3$ LANGSETH_D3 TO/TM
$D_4$ LANGSETH_D4 TO/TM
$D_5$ LANGSETH_D5 TO/TM
Damage $D_C$ LANGSETH_DC TO/TM
$\dot \varepsilon^{pl}_0$ LANGSETH_EPSP0 TO/TM
Room temperature $T_{room}$ LANGSETH_ROOM -
Melting temperature $T_{melt}$ LANGSETH_TMELT -
Plastic strain threshold $\varepsilon^{pl}_D$ LANGSETH_EPL_THRESHOLD -

## GeersContinuousDamage

Damage evolution law following Geers's models. Several laws actually exist, all of the same author, which is why they are gathered in a same class. If the full Geers's model, damage is integrated globally on the structure, and not locally at each integration point. I can give references if needed. All models are based on a characteristic quantity, $\kappa$.

#### Parameters common to all models

Name Metafor Code Dependency
Initiation value $\kappa_i$ GEERS_KAPPA_I TO/TM
Critical value $\kappa_c$ GEERS_KAPPA_C TO/TM

### PowGeersContinuousDamage

Power law. $\kappa$ is the equivalent plastic strain $\varepsilon^{pl}$:

$$D = 1 - \left(\dfrac{\kappa_i}{\kappa}\right)^{n_1} \left(\dfrac{\kappa-\kappa_i}{\kappa_c-\kappa_i}\right)^{n_2} \mbox{ if } \kappa_i\leq\kappa\leq\kappa_c$$

Name Metafor Code Dependency
$n_1$ GEERS_N1 TO/TM
$n_2$ GEERS_N2 TO/TM

### ExpGeersContinuousDamage

Exponential law. $\kappa$ is the equivalent plastic strain $\bar\varepsilon^{pl}$

$$D = 1 - \exp\left(-\beta\left(\kappa-\kappa_i\right)\right)$$

Name Metafor Code Dependency
$\beta$ GEERS_BETA TO/TM

### TanhGeersContinuousDamage

Hyperbolic tangent. $\kappa$ is the equivalent plastic strain $\varepsilon^{pl}$

$$D = \dfrac{1}{2\tanh\left(3\right)} \left(\tanh\left(6\dfrac{\kappa-\kappa_i}{\kappa_c-\kappa_i}-3\right)+\tanh\left(3\right)\right)$$

Name Metafor Code Dependency
Initiation value $\kappa_i$ GEERS_KAPPA_I TO/TM
Critical value $\kappa_c$ GEERS_KAPPA_C TO/TM

### LinGeersContinuousDamage

Law linear. $\kappa$ is a function of the stress triaxiality and the equivalent plastic strain $\varepsilon^{pl}$

$$\dot{\kappa} = C\left<1+A\dfrac{p}{\bar\sigma}\right> \left(\varepsilon^{pl}\right)^B \dot\varepsilon^{pl}$$ where $p$ is the pressure, and $\overline{\sigma}$ the Von Mises stress. $\langle .\rangle$ are Macaulay symbols( $\langle \alpha\rangle = \alpha$ if $\alpha \ge 0$ and $0$ otherwise)

$$\dot D = \dfrac{\dot\kappa}{\kappa_c-\kappa_i}$$

Name Metafor Code Dependency
$A$ GEERS_A TO/TM
$B$ GEERS_B TO/TM
$C$ GEERS_C TO/TM

## References

[1] Lemaitre J. A continuous damage mechanics model for ductile fracture. Journal of Engineering Materials and Technology 1985;107:9–83.

[2] Chaboche JL. Description thermodynamique et phénoménologique de la viscoélasticité cyclique avec endommagement. PhD Thesis, Université Pierre et Marie Curie, Paris VI, 1978.

[3]

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