Figure 4 Representation of the “components” with indication of the two
groups of elements constituting the two passes A; i-th red components of
first pass, i-th blue components of second pass B.
Thermal analysis
The thermal analysis of the welding process is essentially a
mathematical solution of the differential problem based on the equation
of energy conservation:
\(\text{ρC}\frac{\partial T}{\partial t}={\dot{u}}^{{}^{\prime\prime\prime}}+\frac{\partial}{\partial x}\left(k_{x}\frac{\partial T}{\partial x}\right)+\frac{\partial}{\partial y}\left(k_{y}\frac{\partial T}{\partial y}\right)+\frac{\partial}{\partial z}\left(k_{z}\frac{\partial T}{\partial z}\right)\),
(2)
where the temperatures distribution T(x, y, z, t) of the welded
plate is a function of both spatial and time coordinates; ρ ,C and k are the density, the specific heat and the thermal
conductivity of the material, respectively, and \({\dot{u}}^{{}^{\prime\prime\prime}}\) is
the change rate of internal energy per volume unit. Eq. (2) is a
non-linear differential equation since ρ , C and kdepend on the temperature. Initial and boundary conditions of the
problem are respectively:
\(T\left(x,y,z;t=0\right)=T_{0}\), (3)
\({\dot{q}}_{n}\left(x,y,z;t\right)=-\left(k_{x}\frac{\partial T}{\partial x}n_{x}+k_{y}\frac{\partial T}{\partial y}n_{y}+k_{z}\frac{\partial T}{\partial z}n_{z}\right)\),
(4)
where T0 = 23.5 °C is the initial temperature of
the material; \({\dot{q}}_{n}\) is the heat flux at a generic boundary
having an outward local unit vector \(\hat{n}\left(x,y,z\right)\). In
welding problems, at external surfaces, the heat flux \({\dot{q}}_{n}\)may consist of one or more of the following modes: convective heat loss,
radiative heat loss and boundary heat \(\dot{q_{0}}\). The latter has
been neglected in the proposed FE model. Convective and radiative heat
losses on the external surfaces of the welded plates are given
respectively by:
\({\dot{q}}_{\text{nc}}=h_{c}\left[T\left(x,y,z;t\right)-T_{\infty}\right]\),
(5)
\({\dot{q}}_{\text{nr}}=\text{εσ}\left\{\left[T\left(x,y,z;t\right)-T_{\text{az}}\right]^{4}-\left(T_{r}-T_{\text{az}}\right)^{4}\right\}=h_{r}\left[T\left(x,y,z;t\right)-T_{r}\right]\),
(6)
where T∞ and Tr are
respectively the temperatures of the environment transferring heat by
convection and radiation and they are usually equal to the room
temperature; ε is the surface emissivity; σ = 5.67 ·
10-8 W/m2K4 is the
Stefan-Boltzmann constant; hc is the temperature
dependent convective film coefficient and Taz =
-273.15 °C is the absolute zero of the thermal scale used for this work
(Celsius degrees). From Eq. (6) the radiative heat losses can be
expressed in the form of convective heat losses by means of temperature
dependent convective film coefficient hr ,
therefore from Eq. (5) and (6) a unique temperature dependent film
coefficient, H, can be considered:
H = hc + hr . (7)
Particularly important in the thermal model is the heat input per mmQ, reported in Table 3. This is the energy supplied by the
welding machine per unit of length. In the proposed simulation a half of
this energy has been supplied to a half of the seam because one plate
only has been modelled. Therefore, energy supplied to the entire half
welding seam during the simulation is equal to:
\(Q_{\text{real}}=\frac{Q\bullet\ L_{\text{seam}}}{2}\ \), (8)
where: Lseam is the length of welding bead.
This energy can be subdivided into three parts:
sensible heat : energy to heat the weld material from the
initial temperature (T0 ) to the solidus
temperature (Ts ):
\(Q_{\text{sensible}}=\text{vol}_{\text{seam}}\bullet\int_{T_{0}}^{T_{S}}\text{ρ\ C\ dT}=m_{\text{seam}}\int_{T_{0}}^{T_{S}}\text{C\ dT}\),
(9)
where: ρ and C are the density and specific heat of the
material respectively and volseam andmseam are the volume and the mass of half welding
bead;
latent heat : energy due to phase transition from the solidus
temperature (TS ) to the liquidus temperature
(TL ):
\(Q_{\text{latent}}=m_{\text{seam}}\ \bullet q_{\text{latent}}\) ,
(10)
where: mseam is the mass of half welding bead andqlatent is the latent heat per mass unit;
the energy to further heat the weld material is equal to:
\(Q_{\text{body\ flux}}=Q_{\text{real}}-Q_{\text{sensible}}-Q_{\text{latent}}\), (11)
while the energy to be applied to the single components is:\(Q_{\text{component}}=\frac{Q_{\text{body\ flux}}}{n_{\text{component}}}\), (12)
where: ncomponent is the number of components of
whole half welding bead.
This latter part of the energy acts as volumetric generation of the
internal energy \({\dot{u}}^{{}^{\prime\prime\prime}}\) (Table 3) and it is computable by
Eq. 13:\({\dot{u}}^{{}^{\prime\prime\prime}}=\frac{Q_{\text{component}}}{\text{vol}_{\text{component}}\ \bullet t_{\text{weld}}}=\frac{Q_{\text{body\ flux}}}{n_{\text{component}}}\bullet\ \frac{v}{\text{vol}_{\text{component}}\ \bullet L_{\text{component}}}=\frac{Q_{\text{body\ flux}}\ \bullet v}{\text{vol}_{\text{seam}}\bullet\ L_{\text{component}}}\), (13)
where: volcomponent andLcomponent are the volume and the length of the
single component, respectively; v is the welding speed andtweld is the time necessary to travel a distance
equals to the length of the single component by Eq. (14);\(t_{\text{weld}}=\frac{L_{\text{component}}}{v}\ \). (14)
In the proposed FE model, the specific power \({\dot{u}}^{{}^{\prime\prime\prime}}\) has
been applied to each component during the timetweld as volumetric flux and it has been applied
by means of the law shown in Figure 5.
The load has been applied so that the area under the load curve is
constant and equals to \({\dot{u}}^{{}^{\prime\prime\prime}}\) at varying travel time
(tweld ). Two ramps [with duration of 0.5% oftweld ], to avoid the discontinuity during the
load application, and a little time offset of 2·10-6s between two load curves have been defined, in order to
encourage the convergence of the solution. The height of the trapeziumh is computable by Eq. (15):
\({\dot{u}}^{{}^{\prime\prime\prime}}=A_{\text{trapezium}}=\frac{\left(B+b\right)\bullet h}{2}=\frac{\left\{\left(1\bullet t_{\text{weld}}\right)+\left[\left(1-2\bullet 0.005\right)\ t_{\text{weld}}\right]\right\}\ \bullet h}{2}=\left(1-0.005\right)\bullet\ t_{\text{weld}}\ \bullet h\)(15)
and hence:
\(h=\frac{{\dot{u}}^{{}^{\prime\prime\prime}}}{\left(1-0.005\right)\bullet t_{\text{weld}}}\).
(16)