1. Reference problem#
1.1. Geometry#
We consider a structure with a solder composed of 19 beads.
The calculation is limited to the successive addition of the first 2 cords.
1.2. Material properties#
The thermo-elastic properties of the material are defined as a function of temperature.
Thermal parameters:
Conductivity \(\mathrm{\lambda }(W\mathrm{.}{\mathit{mm}}^{-1}°{K}^{-1})\):
T (°C) |
\(\lambda\) |
0.014 |
|
0.0158 |
|
0.0172 |
|
0.0186 |
|
0.02 |
|
0.0211 |
|
0.0222 |
|
0.0232 |
|
0.0241 |
|
0.0248 |
|
0.0255 |
|
0.0269 |
|
0.0283 |
|
0.283 |
Table 1.2-1 : Thermal conductivity
Volume heat capacity \(\mathrm{\rho }{C}_{p}(J\mathrm{.}{\mathit{mm}}^{-3.}°{K}^{-1})\)
T (°C) |
\(\mathrm{\rho }{C}_{p}\) |
0.0036 |
|
0.0039 |
|
0.0042 |
|
0.0043 |
|
0.0043 |
|
0.0044 |
|
0.0047 |
|
0.0049 |
|
0.0049 |
|
0.0050 |
Table 1.2-2 : Volume heat capacity
Thermo-mechanical parameters:
Thermo-elastic parameters:
Young’s module \(E(\mathit{Pa})\)
Poisson’s ratio: \(\mathrm{\nu }\)
Thermal expansion coefficient \(\alpha\)
Expansion coefficient definition temperature: \({T}_{\mathit{ref}}=20°C\)
Elasticity limit \({\mathrm{\sigma }}_{y}(\mathit{Pa})\)
T (°C) |
\(E(\times {10}^{6})\) |
T (°C) |
T (°C) |
\(\mathrm{\nu }\) |
T (°C) |
\(\alpha (\times {10}^{-6})\) |
T (°C) |
\({\sigma }_{y}(\times {10}^{6})\) |
|||
197,000 |
0.296 |
15.9 |
|||||||||
184,000 |
0.298 |
16.25 |
|||||||||
176,500 |
200 |
200 |
0.304 |
16.70 |
|||||||
168,000 |
0.315 |
17.10 |
|||||||||
160,000 |
0.320 |
17.40 |
|||||||||
151,500 |
0.323 |
17.90 |
|||||||||
142,500 |
0.326 |
18.30 |
|||||||||
130,000 |
0.330 |
18.50 |
|||||||||
108,000 |
0.336 |
18.80 |
|||||||||
81 500 |
0.339 |
19.10 |
|||||||||
20,000 |
0.346 |
19.40 |
|||||||||
3000 |
0.349 |
1100 |
1100 |
19.60 |
|||||||
501 |
0.353 |
19.90 |
|||||||||
501 |
0.353 |
20 |
Table 1.2-3 : Thermal last-ic properties
Thermoplastic parameters:
Linear work hardening laws:
The parameters of the law of restoration of work hardening are: Restoration start temperature \({T}_{\mathit{mini}}=1075°C\)
Full catering temperature \({T}_{\mathit{maxi}}=1150°C\)
Coefficient \(\mathrm{\alpha }\) and \({\mathrm{\tau }}_{\mathrm{\infty }}\) as a function of temperature:
T (°C) |
\(\mathrm{\alpha }\) |
T (°C) |
\({\mathrm{\tau }}_{\mathrm{\infty }}\) |
|
1.154 |
1.0 |
|||
1.335 |
0.877 |
|||
1.308 |
0.767 |
|||
1.165 |
0.682 |
|||
0.181 |
0.576 |
|||
0.687 |
0.071 |
|||
0.408 |
0.052 |
|||
0.045 |
||||
Table 1.2-4 : Properties of the law of hardening restoration
1.3. Boundary conditions and loads#
The thermo-mechanical calculation includes the following calculation steps:
Nonlinear Thermal Calculation:
When each cord is removed, a thermal calculation is carried out where three loads are taken into account:
A thermal source on the added cord. This source is defined by a function that depends on the following process parameters: speed Vs, power Q and efficiency \(\mathrm{\eta }\). The equivalent heat source is defined by a triangle function on the surface of the cord.
Vs_1 = 1.53 mm/s Vs_2 = 1.19 mm/s
Q = 520 and \(\mathrm{\eta }=0.75\)
Radiation and convection on external surfaces with the parameters \({T}_{\mathit{ext}}=20°C\), the emissivity \(\mathrm{ϵ}=0.2\), the Boltzmann constant \(\mathrm{\sigma }=5.67e-14\). The exchange coefficient for convection is \(h=15e-06\). ,
The initial temperature is 20°C throughout the model.
Non-linear mechanical calculation:
The mechanical calculation is carried out in two phases by applying the thermal load with expansion clamping at the ends of the tube. The two phases correspond to the solidification phase (temperature of the cord higher than the melting temperature \({T}_{\mathit{fusion}}=1450°C\)), for the next phase, there is an adjustment of the position and deformation of the cord. The materials are modified between the two calculation phases.
The application of the cord 1 takes place between the moments 0. and 990.
The application of the cord 2 takes place between the moments 1000. and 1990.