Differences

This shows you the differences between two versions of the page.

--- doc:user:elements:volumes:thixo_burgoscohesionmatlaw [2013/07/11 15:16] – created joris
+++ doc:user:elements:volumes:thixo_burgoscohesionmatlaw [2016/03/30 15:23] (current) – external edit 127.0.0.1
@@ Line 1: / Line 1: @@
+====== Cohesion degree======
+The ''CohesionMatLaw'' class manages all cohesion degree evolution laws, specific to thixotropic materials. These laws are described below.
+:!: Careful: Only works if used with thixotropic materials (''ThixoEvpIsoHHypoMaterial'' or  ''ThixoTmEvpIsoHHypoMaterial'').
+===== IsothCohesionMatLaw =====
+=== Description ===
+The evolution of the structural parameter λ can be expressed by a differential equation
+that describes the kinetics between the agglomeration of the solid grains and the destruction
+of the solid bonds due to shearing. Solved using Newton-raphson, this equation is isothermal since it does not take the [[doc:user:elements:volumes:thixo_scheilliquidfractionmatlaw|liquid fraction]] into account.
+$$
+d \lambda / dt = a (1 - \lambda)^{1+e} - b \lambda e^{c \dot{\bar{\epsilon}}^{vp} } (\dot{\bar{\epsilon}}^{vp})^d
+$$
+=== Parameters ===
+^   Name  ^     Metafor Code  ^  Dependency ^
+|  $ a $  |  ''LAMBDA_A''  |      ''TM/TO''     |
+|  $ b $  |  ''LAMBDA_B''  |      ''TM/TO''     |
+|  $ c $  |  ''LAMBDA_C''  |      ''TM/TO''     |
+|  $ d $  |  ''LAMBDA_D''  |      ''TM/TO''     |
+|  $ e $  |  ''LAMBDA_E''  |      ''TM/TO''     |
+===== BurgosCohesionMatLaw =====
+=== Description ===
+Burgos law, this time considering the [[doc:user:elements:volumes:thixo_scheilliquidfractionmatlaw|liquid fraction]]. The cohesion degree is an explicit function of the equivalent plastic strain rate (integration over a time step where the equivalent plastic strain is supposed to remain constant).
+$$
+\lambda =\lambda_e + ( \lambda_0 - \lambda_e)  e^{F(\lambda) \Delta t}
+$$
+where
+$$
+F(\lambda) = -\left(  a'+b' e^{c \dot{\bar{\epsilon}}^{vp}} (\dot{\bar{\epsilon}}^{vp})^{d'}  \right)
+$$
+$$
+\lambda_e = \frac{-a'}{F(\lambda)}
+$$
+$$
+a' = a (1-f_l) + f e^{-g f_l}
+$$
+$$
+b' = b f_l + f e^{-g (1-f_l)}
+$$
+$$
+d' = d (1-(f_l)^{e})
+$$
+=== Parameters ===
+^   Name                 ^     Metafor Code     ^  Dependency ^
+|  $ a $  |  ''LAMBDA_A''  |      ''TM/TO''     |
+|  $ b $  |  ''LAMBDA_B''  |      ''TM/TO''     |
+|  $ c $  |  ''LAMBDA_C''  |      ''TM/TO''     |
+|  $ d $  |  ''LAMBDA_D''  |      ''TM/TO''     |
+|  $ e $  |  ''LAMBDA_E''  |      ''  /  ''     |
+|  $ f $  |  ''LAMBDA_F''  |      ''  /  ''     |
+|  $ g $  |  ''LAMBDA_G''  |      ''  /  ''     |
+===== FavierCohesionMatLaw =====
+=== Description ===
+Burgos law, this time considering the [[doc:user:elements:volumes:thixo_scheilliquidfractionmatlaw|liquid fraction]]. The cohesion degree is an explicit function of the equivalent plastic strain rate (integration over a time step where the equivalent plastic strain is supposed to remain constant). Percolation is also taken into account, meaning that the cohesion degree approaches zero when the liquid fraction approaches a critical value $ f_c = e $.
+$$
+\lambda = \lambda_e + ( \lambda_0 - \lambda_e)  e^{F(\lambda) \Delta t} \mbox{ if } f_l < f_c = e
+$$
+$$
+\lambda = 0 \mbox{ if } f_l \geq f_c = e
+$$
+where
+$$
+F(\lambda) = -\left( a'+b' e^{c \dot{\bar{\epsilon}}^{vp}} (\dot{\bar{\epsilon}}^{vp})^d \right)
+$$
+$$
+\lambda_e = \frac{-a'}{F(\lambda)}
+$$
+$$
+a' = a (1-f_l) + f e^{-g f_l}
+$$
+$$
+b' = b f_l + f e^{-g (1-f_l)}
+$$
+=== Parameters ===
+^   Name  ^     Metafor Code     ^  Dependency ^
+|  $ a $  |  ''LAMBDA_A''  |      ''TM/TO''     |
+|  $ b $  |  ''LAMBDA_B''  |      ''TM/TO''     |
+|  $ c $  |  ''LAMBDA_C''  |      ''TM/TO''     |
+|  $ d $  |  ''LAMBDA_D''  |      ''TM/TO''     |
+|  $ e $  |  ''LAMBDA_E''  |      ''  /  ''     |
+|  $ f $  |  ''LAMBDA_F''  |      ''  /  ''     |
+|  $ g $  |  ''LAMBDA_G''  |      ''  /  ''     |