**2.4 Finite element modeling**

*C*∗

*New Challenges in Residual Stress Measurements and Evaluation*

*km*ð Þ¼ *T*

ture, and *T* and *k T*ð Þ maintain their previous definitions.

*2.3.6 Combined boundary conditions*

tive heat transfer were given by [19] as

"FILM" is written to simulate heat loss.

*Latent heat of fusion for stainless steel 304.*

**Table 1.**

**92**

(

The boundary conditions shown in Eq. (4) can be rewritten as

*hr* <sup>¼</sup> *εσ <sup>T</sup>*<sup>2</sup> <sup>þ</sup> *<sup>T</sup>*<sup>2</sup>

0

loss and when temperature is high, radiation becomes dominant. As shown in Eq. (10), radiation coefficient is the third-order function of temperature *T*, which is highly nonlinear. This would greatly increase the computational expense and time. Based on experimental data, an empirical formula combining convective and radia-

�

where *hr* is the radiation coefficient expressed as

*<sup>h</sup>* <sup>¼</sup> *hc* <sup>þ</sup> *εσ <sup>T</sup>*<sup>2</sup> <sup>þ</sup> *<sup>T</sup>*<sup>2</sup>

where *C*<sup>∗</sup>

*2.3.5 Marangoni effect*

*<sup>p</sup>* ð Þ¼ *<sup>T</sup> Cp*ð Þþ *<sup>T</sup> <sup>L</sup>*

dependent-specific heat, *L* is the latent heat of fusion, *Tm* is the melting temperature, and *T*<sup>0</sup> is the ambient temperature. The values of the latent heat of the fusion, solidus temperature, and liquidus temperature of SS 304 [16] appear in **Table 1**.

As discussed in [17], the temperature distribution is significantly impacted by the effect of Marangoni flow, which is caused by the thermocapillary phenomenon. To obtain an accurate thermal field solution, based on the method proposed by [18], an artificial thermal conductivity was used to account for the Marangoni effect:

where *km*ð Þ *T* is the modified thermal conductivity, *Tliq* is the liquidus tempera-

*<sup>K</sup>*ð Þj *<sup>Δ</sup><sup>T</sup>* � *<sup>n</sup>* <sup>Γ</sup> <sup>¼</sup> ½ð Þ �*hc* � *hr* ð Þ *<sup>T</sup>* � *<sup>T</sup>*<sup>0</sup> �j<sup>Γ</sup> <sup>Γ</sup> <sup>∉</sup> <sup>Λ</sup>

0

� �ð Þ *<sup>T</sup>* <sup>þ</sup> *<sup>T</sup>*<sup>0</sup> <sup>≈</sup> <sup>2</sup>*:*<sup>41</sup> � <sup>10</sup>�<sup>3</sup>

where *h* is the combined heat transfer coefficient which is a lower-order function of temperature *T* compared with *hr*. The associated loss in accuracy using this relationship is estimated to be less than 5% [20]. In ABAQUS, a user subroutine

**Latent heat of fusion (J/kg) Solidus temperature (K) Liquidus temperature (K)**

273,790 1703 1733

Eq. (9) shows that when the temperature is low, convection is dominant in heat

½*Q* � �ð Þ *hc* � *hr* ð Þ *T* � *T*<sup>0</sup> �j<sup>Γ</sup> Γ∈Λ

� �ð Þ *<sup>T</sup>* <sup>þ</sup> *<sup>T</sup>*<sup>0</sup> (10)

*k T*ð Þ *T* ≤*Tliq* 2*:*5 � *k T*ð Þ *T*>*Tliq*

*Tm* � *T*<sup>0</sup>

*<sup>p</sup>* ð Þ *T* is the modified specific heat, *Cp*ð Þ *T* is the original temperature-

(7)

(8)

(9)

*εT*<sup>1</sup>*:*<sup>61</sup> (11)
