====== RACO ======
===== Description =====
Constitutive law for convection and radiation for elements CONRA and [[elements:cora3|CORA3]]. \\
Last revision: L. Grisard - sept 1990

==== The model ====
Non linear thermal analysis of solids. This constitutive law takes account of heat transfer between the solid and the external world by convection and radiation. The coefficients of convection and radiation may be defined as functions of temperature.
==== Files ====
Prepro: LRACO.F \\
Lagamine: RACO1.F
==== Subroutines ====
^File^Subroutine^Description^
|RACO1.F| RACO1|Main subroutine of the law |
===== Availability =====
|Plane stress state| YES |
|Plane strain state| YES |
|Axisymmetric state| YES |
|3D state| YES |
|Generalized plane state| YES |
===== Input file =====

==== Parameters defining the type of constitutive law ====
^ Line 1 (2I5, 60A1)^^
|IL|Law number|
|ITYPE| 120|
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|
==== Integer parameters ====
^ 1 line (I5) ^^
|NTEMP|Number of temperatures at which material data are given|

==== Real parameters ====
^ 1 line repeated NTEMP times (3G10.0) ^^
|T| Temperature|
|CONVEC|Convection coefficient at temperature T (h)|
|RADIA| Radiation coefficient at temperature T (σε with σ=Boltzmann's constant; ε=emissivity)|

=== Remarks about emissivity ε ===
RADIA = $\sigma\varepsilon$ but sometimes $\varepsilon -> 1-\varepsilon$
|{{  :laws:raco_1.png?400|}}| \[\frac{2\varepsilon}{2-\varepsilon}\] $\varepsilon$ no RAYON (ECHRA) elements |
|{{ :laws:raco_2.png?400 |}}| \[\frac{1}{\frac{1-\varepsilon}{\varepsilon}+1}=\frac{\varepsilon}{1-\varepsilon+\varepsilon}=\varepsilon\] with RAYON (ECHRA) elements (leak flow)|
|{{ :laws:raco_3.png?400 |}}|\[\frac{1}{2\frac{1-\varepsilon}{\varepsilon}+1}=\frac{\varepsilon}{2-2\varepsilon+\varepsilon}=\frac{\varepsilon}{2-\varepsilon}\] Between 2 surfaces: \[\varepsilon_{12}^*=\frac{\varepsilon_1\varepsilon_2}{\varepsilon_1+\varepsilon_2-\varepsilon_1\varepsilon_2}\] \[\text{If }\varepsilon_1=\varepsilon_2=\varepsilon => \varepsilon_{12}^*=\frac{\varepsilon^2}{2\varepsilon-\varepsilon^2}=\frac{\varepsilon}{2-\varepsilon}\] with unique RAYON (ECHRA) element -> view factor = 1|
|{{ :laws:raco_4.png?400 |}}|\[\frac{1}{2\frac{1-\varepsilon}{2\varepsilon}}=\frac{\varepsilon}{2-\varepsilon}\] $\varepsilon_1=\varepsilon_2$ without RAYON (ECHRA) element|
|{{ :laws:raco_5.png?600 |}}__General case__: \[q_{pi}=\frac{A_i\varepsilon_i}{1-\varepsilon_i}\sigma(T_i^{*4}-T_p^4)\]  \[q_{ij}=\sigma A_i F_{ij}(T_j^{*4}-T_i^{*4})\] With $A_i F_{ij}$ computed by ECHRA \\ $T^*$ = Fictitious temperature taking into account the computation of radiosity ||
===== Stresses =====
==== Number of stresses ====
1
==== Meaning ====
|SIG(NPI)|Heat flow exchanged by convection and radiation|

===== State variables =====
==== Number of state variables ====
3
==== List of state variables ====
|Q(1)|Temperature of the solid at the point under consideration ($= T$)|
|Q(2)|Temperature of the external world ($= T_a$)|
|Q(3)|$=h+\sigma \varepsilon \left(T^3_a+T^2_a*T+T_a*T^2+T^3\right)$|