1 Introduction

Amongst the diverse range of nuclear power generation technologies, the superior fuel economy of fast breeder reactors (FBRs) that is made possible by their use of fast neutrons to breed plutonium-239 from uranium-238 makes them well known as the most advanced reactor type. The commercialisation of FBR plants has therefore been regarded as one of the most effective energy sources to replace the gradually diminishing fossil fuels that are used at present. Austenitic stainless steels have been widely used in various industrial fields such as the oil and gas industry, as well as nuclear power plants, mainly because of their superior corrosion resistance and mechanical properties under demanding service environments. An advanced type of stainless steel (316FR), which offers an improved creep fatigue resistance compared to other austenitic stainless steels, is hoped to be used as a structural material in the primary coolant circuit of next-generation commercial FBRs [1]. However, when FBR plants have been in operation for a long time (max. about 40 years), then components suffering from the effects of sigma phase embrittlement, creep fatigue or corrosion in a liquid sodium environment will require essential maintenance, which typically involves welding [2, 3].

Before establishing a repair welding process for aging FBR plants, one must first consider the main differences between them and other common types of nuclear reactor (e.g. light water-cooled reactors). For instance, the high thermal conductivity of FBRs means that they use molten sodium metal as a coolant instead of water, and their high degree of heat generation also means they operate at a much higher temperature of 773–823 K. After long-term service, corrosion induced by the flow of liquid sodium at these high service temperatures can damage the surface layer of the structural materials [1–3]. Consequently, a laser beam weld overlay has been considered a promising candidate for the repair of aging FBR plants, as it offers several advantages over laser beam welding (LBW) from both a metallurgical perspective (reduced segregation on solidification, a refinement of microstructure and a minimisation of the heat-affected zone) and manufacturing perspective (production lines can be highly automated and flexible).

During any welding process for austenitic stainless steels, special attention needs to be paid to the alloy’s weldability, which is commonly expressed in terms of its hot cracking [4–18]. Previous reports have suggested that laser welds are more susceptible to solidification cracking than arc welds in several kinds of stainless steels [6, 8, 10], and so in order to achieve sound repair welds in aging FBRs, there is a need for a more detailed examination of the solidification cracking behaviour of laser welds. The present study therefore uses a Varestraint testing setup combined with a LBW apparatus (LBW transverse-Varestraint testing [19, 20]) to quantitatively evaluate the solidification cracking susceptibility (i.e. the solidification brittle temperature range, BTR) of laser welds in type 316FR stainless steel. Particular focus is given to what effect the solidification mode (primary austenite–secondary ferrite, AF mode; and primary δ-ferrite–secondary austenite, FA mode) has on the BTR behaviour with various welding speeds and two kinds of filler metal. These experimental results are used to discuss the variation in BTR with welding speed through theoretical analysis of the temperature range over which solid and liquid phases coexist during non-equilibrium solidification, taking into account dendrite supercooling, diffusion and segregation on solidification.

2 Materials and experimental procedure

2.1 Materials

The materials used in this study were 100 mm (length) × 50 (width) × 5 (thickness) plates made from two kinds of filler metal (316FR-A, 316FR-B), whose compositions lie within the range of type 316FR stainless steel [1, 21]. The chemical compositions (mass%) of each filler metal is shown in Table 1. The Creq/Nieq value indicates that 316FR-A exhibits AF mode solidification, whereas 316FR-B experiences FA mode solidification [12, 13].

Table 1 Chemical composition of the steels used (mass%)
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2.2 Laser beam welding transverse-Varestraint test

The LBW transverse-Varestraint testing apparatus used is shown in Fig. 1 and consisted of a LBW robot combined with a Varestraint testing apparatus. During testing, a hydraulic pressure was applied immediately after transverse welding, with the entire testing sequence being imaged through a protective filter using a high-speed camera to closely control the timing of the testing sequence, i.e. to maintain the consistency between, and to remove the unavoidable time lag between procedures such as the yoke-drop and the extinguishment of the laser beam. The test parameters are listed in Table 2. Two levels of welding speed (20.0 and 40.0 mm/s) were used, the laser power being adjusted to obtain a half-penetration bead at each speed. Argon was used as a shield gas during welding to protect the bead surface from oxidation, with the augmented strain being varied from between 0.25 and 5.88 %. Figure 1 also shows the dimensions of the test specimen used (100 mm × 50 mm × 5 mm) together with the bending block arrangement. The length of the transverse weld bead was adjusted to 30 mm for each welding speed, and the temperature history during solidification was measured using a thermocouple (W-Re, diameter 0.6 mm) plunged directly into the weld bead during LBW (and under the same welding conditions for transverse welding) to obtain the cracking temperature range. The surface of cracks formed during transverse-Varestraint testing was observed by scanning electron microscope (SEM). Further details of the LBW transverse-Varestraint testing procedure used can be found in the author’s previous studies [19, 20].

Fig. 1
figure 1

Appearance and schematic descriptions for LBW transverse-Varestraint test

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Table 2 Parameters used for the LBW transverse-Varestraint test
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2.3 Gas tungsten arc welding transverse-Varestraint test

Transverse-Varestraint testing using gas tungsten arc welding (GTAW transverse-Varestraint test) was also conducted using the test conditions listed in Table 3. The dimensions of the specimens used were identical to those in the LBW transverse-Varestraint test (displayed in Fig. 1). The temperature history during solidification was measured using a thermocouple (Pt-12%Rh/Pt) plunged directly into the weld bead under the same conditions for transverse welding to obtain the cracking temperature range. The crack surface produced by the transverse-Varestraint test was also observed by SEM.

Table 3 Parameters used for the GTAW transverse-Varestraint test
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3 Solidification cracking behaviour during laser beam welding

3.1 Appearance and characterization of solidification cracks

Figure 2 shows typical examples of the solidification cracks produced by LBW transverse-Varestraint testing at a welding speed of 20.0 mm/s for 316FR-B. These cracks occurred within the central region of the weld bead for both steels and were classified as either a centreline or transverse crack. Figure 2 also shows the fracture surface of a crack that had the maximum length. The fracture surfaces in Fig. 2 indicate shape of centreline fracture (enlarged a and b) with a totally dendritic structure with a trace of melting (enlarged c). In other words, the cracks formed by LBW transverse-Varestraint testing can be regarded as solidification cracks, with similar results obtained in the case of 316FR-A.

Fig. 2
figure 2

Appearance and fracture surface (SEM) after LBW transverse-Varestraint test

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3.2 Susceptibility to solidification cracking

Figure 3 provides a summary of the solidification cracking susceptibilities, as evaluated from the crack length and number of cracks observed by SEM after transverse-Varestraint testing (316FR-B, welding speed 20.0 mm/s). Generally, the solidification cracking behaviour in the transverse-Varestraint test can be classified into three types, namely, crack, back-fill (filled with liquid from molten pool after crack initiation) and blank (distance from fusion line to crack initiation point) from the fusion line. Each bar in Fig. 3 represents an individual crack, which is classified as either a crack, back-fill or blank. As can be seen, the cracking susceptibility (number and length of cracks) increases with an increase in the applied strain to 2.44 % of augmented strain, and the cracking susceptibility saturated at 3.45 % strain level. In addition, a solidification brittle temperature range (BTR) of 29 K was obtained for 316FR-B at 20.0 mm/s by using the maximum crack length and the cooling rate measured by the thermocouple. This BTR is also shown in the ductility curve for solidification cracking in Fig. 4. The procedure by which this BTR value was converted can be found in the author’s previous studies [19, 20].

Fig. 3
figure 3

Location and length of the solidification crack formed during the LBW transverse-Varestraint test

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Fig. 4
figure 4

High-temperature ductility curve of solidification cracking and BTR obtained from the LBW transverse-Varestraint test

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3.3 Change in solidification brittle temperature range with welding speed

The variation in BTR behaviour for both 316FR-A and 316FR-B is presented in Fig. 5 as a function of the welding speed from 1.67 mm/s (GTAW transverse-Varestraint) to 40.0 mm/s (LBW transverse-Varestraint test). This shows that the BTR increased from 37 to 46 K in the case of 316FR-A and from 14 to 40 K with 316FR-B. It therefore follows that the solidification cracking susceptibility of 316FR-A and 316FR-B steels increased through LBW. The greater increase in BTR with 316FR-B (26 K) than 316FR-A (9 K) confirms that its relation to welding speed depends on the whether the solidification mode is AF or FA. Based on this, the governing factor behind the variation in BTR for type 316FR stainless steel was explored through theoretical calculation of the temperature range at which solid and liquid phases coexist during solidification. Especially, the BTR’s level is extremely lower than common austenitic stainless steels (for instance type 310S stainless steel) [20], because type 316FR stainless steels have been specially developed for the future FBRs, optimizing impurity element such as phosphorous (P), suppressing solidification cracking susceptibility [3]. And, author already clearly shows that the BTR shrank from 146 to 120 K (26 K reduction) with an increase in the welding speed from 1.67 mm/s (GTAW) to 40.0 mm/s (LBW) as reported by previous study for type 310S stainless steel [20]. In other words, variation tendency of solidification cracking susceptibility (BTR) is different in accordance with solidification mode of austenitic stainless steels.

Fig. 5
figure 5

Relationship between welding speed and BTR

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4 Mechanism behind the variation in solidification brittle temperature range during laser beam welding

Hereinafter, the mechanism by which the BTR varies with welding speed (i.e. the differing extent to which it is increased) is discussed by employing a numerical calculation of the temperature range of solid/liquid coexistence. The BTR is generally characterised by its upper and lower temperature limits; the former corresponds to the solidification initiation temperature (T I ), while the latter is approximated by the solidification completion temperature. During the welding process, however, dendritic growth is accompanied by a segregation of solute elements into the liquid phase. This increasing concentration of elements in the liquid during solidification causes the true solidification completion temperature (T C ) to deviate from the nominal solidus temperature ((T S )). The difference between the solidification initiation (T I ) and completion (T C ) temperature is regarded as the temperature range over which solid and liquid coexist during welding solidification, and any mechanism behind a change in this value as a function of welding speed would govern the BTR change mechanism. As such, the change in solidification initiation (T I ) and completion (T C ) temperature was computed using the Kurz-Giovanola-Trivedi (KGT) model [22, 23] and a solidification segregation model, respectively.

4.1 Theoretical basis for the calculation of solidification initiation temperature (T I )

The solidification initiation temperature (T I ) can be approximated from the dendrite tip temperature (T Tip ), for which a KGT model of the Fe-Cr-Ni ternary system was applied [5, 11, 24, 25]. The calculation procedure is similar to that used in several previously reported studies [5, 11, 24, 25], but a brief description of the theoretical basis is provided below.

The solidification initiation temperature (T I ) (dendrite tip temperature (T Tip )) can be deduced from the difference between the equilibrium liquidus temperature (T eq L ) and actual undercooling degree (ΔT) as follows:

$$ {T}_{Tip}\ \left(={T}_I\right)={T}_L^{eq}-\varDelta T $$
(1)
$$ \varDelta T=\varDelta {T}_C+\varDelta {T}_R+\varDelta {T}_K+\varDelta {T}_{cell} $$
(2)

where ΔT C is the constitutional undercooling, ΔT R is the curvature undercooling, ΔT K is the kinetic undercooling, and ΔT cell is the cellular undercooling.

Constitutional undercooling (ΔT C ) is defined as

$$ \varDelta {T}_C={m}_i^{eq}{C}_i^0-{m}_i^V{C}_i^L $$
(3)

where m eq i is the equilibrium liquidus gradient of each alloying element, m V i is the velocity-dependent liquidus gradient, C 0 i is the initial composition of alloying elements, and C L i is the liquid concentration at the dendrite tip. Since equilibrium partitioning cannot be assumed in a rapid solidification process like that which follows welding, a theory that considers the deviation from the equilibrium partitioning discussed by Boettinger et al. [26] and the variation in partition coefficient with growth velocity suggested by Aziz [27] was introduced:

$$ {m}_i^V={m}_i^{eq}\left\{1+\frac{k_i^{eq}-{k}_i^V\left(1- \ln \frac{k_i^V}{k_i^{eq}}\right)}{1-{k}_i^{eq}}\right\} $$
(4)
$$ {k}_i^V=\frac{k_i^{eq}+\frac{a_0V}{D_i^L}}{1+\frac{a_0V}{D_i^L}} $$
(5)

where V is the dendrite growth velocity, k V i is the velocity-dependent partitioning coefficient, k eq i is the equilibrium partitioning coefficient, D L i is the temperature-dependent diffusion coefficient of the alloying elements in the liquid, and a 0 is the length scale related to the interatomic distance. Accordingly, the stability criterion, C L i , is determined by the following equations:

$$ {C}_i^L=\frac{C_i^0}{1-\left(1-{k}_i^V\right){I}_V\left({P}_i^C\right)} $$
(6)
$$ {I}_V\left({P}_i^C\right)={P}_i^C \exp \left({P}_i^C\right){\displaystyle {\int}_{P_i^C}^{\infty}\frac{exp\left(-z\right)}{z}dz} $$
(7)
$$ G{R}^2+2R{\displaystyle \sum_i}{m}_i^V{P}_i^C\left(1-{k}_i^V\right){C}_i^L{\xi}_i^C+4{\pi}^2\varGamma =0 $$
(8)

where P C i is the Peclet number of the solute, I V (P C i ) is Ivantsov’s solution, G is the temperature gradient, V′ is the welding velocity, ε is the cooling rate, and ξ C i is the absolute stability coefficient:

$$ {\xi}_i^C\left\{\begin{array}{cc}\hfill 1-\frac{2{k}_i^V}{\sqrt{1+{\left(\frac{2\pi }{P_i^C}\right)}^2}-1+2{k}_i^V}\hfill & \hfill \cdots \left({P}_i^C\le \frac{\pi^2}{\sqrt{k_{V,i}}}\right)\hfill \\ {}\hfill \frac{\pi^2}{k_i^V{P}_i^C}\hfill & \hfill \cdots \left({P}_i^C>\frac{\pi^2}{\sqrt{k_{V,i}}}\right)\hfill \end{array}\right. $$
(9)

The curvature undercooling (ΔT R ) is expressed as [27, 28]

$$ \varDelta {T}_R=-\frac{2\varGamma }{R} $$
(10)

where Γ is the Gibbs-Thompson parameter.

The kinetic under cooling (ΔT K ) can be described as [11, 25]

$$ \varDelta {T}_K=\frac{V}{\mu_0}=\frac{R_g{T}_m^2}{\varDelta {H}_f{V}_0}V $$
(11)

where μ 0 is the linear interface kinetic coefficient, V 0 is the sonic velocity in the liquid, ΔH f is the enthalpy of fusion, and T m is the liquidus temperature of pure iron.

Burden et al. defined ΔT cell as [11, 25, 29]

$$ \varDelta {T}_{cell}=\frac{GD}{V} $$
(12)

From this theoretical basis, the solidification initiation temperature (T I ) was determined by first calculating the dendrite tip temperature (T Tip ) for the δ-ferrite and austenite phase in both 316FR-A and 316FR-B, and then selecting the highest value as the solidification initiation temperature (T I ). To calculate the constitutional undercooling component (ΔT C ), the binary system for each alloying element was used (i.e. Fe-Cr, Ni, Mo, Si, Mn, P, S, N and C), and the contribution of each system was combined as

$$ {T}_I={T}_L^{eq}+{\displaystyle \sum_i}{m}_i^{eq}{C}_i^0\left\{1-\frac{1+\frac{k_i^{eq}-{k}_i^V\left(1- \ln \frac{k_i^V}{k_i^{eq}}\right)}{1-{k}_i^{eq}}}{1-\left(1-{k}_i^V\right){I}_V\left({P}_i^C\right)}\right\}-\frac{2\varGamma }{R}-\frac{R_g{T}_m^2}{\varDelta {H}_f{V}_0}V-\frac{GD}{V} $$
(13)

4.2 Theoretical basis for the calculation of solidification completion temperature (T C )

In order to calculate the solidification completion temperature (T C ), the solidification segregation behaviour was calculated for each binary system (i.e. Fe-Cr, Ni, Mo, Si, Mn, P, S, N and C) using the finite differential method with the divorced-eutectic solidification model shown in Fig. 6. This is based on [29] and assumes the cross-sectional shape of the dendrites to be essentially a hexagonal prism and applies a one-dimensional half-quadrangle model. For 316FR-A (AF mode solidification), austenite solidifies from the cell core to the boundary as a primary phase, which is then followed by an δ-ferrite secondary phase that solidifies from the cell boundary to the cell core (and vice-versa for the FA mode solidification of 316FR-B). This means that the residual liquid phase at the completion of solidification is located between the austenite and δ-ferrite phases in both 316FR-A and 316FR-B. The principle behind this calculation procedure is similar to that previously reported [11, 30–32], but provided below is a brief description of this segregation model.

Fig. 6
figure 6

Schematic illustration of the analysis model (solidification segregation model) for solidification completion temperature (*δ: δ-ferrite, γ: austenite)

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The distribution of solute elements during solidification was determined by solving the diffusion equation. For this, symmetrical boundary conditions were applied to both end segments, with the final segment of the liquid phase being defined as the solidification boundary (i.e. the dendrite cell boundary). The diffusive flux J i from any given segment i to the next i + 1 is given by Fick’s first law:

$$ {J}_i=D\frac{C_{i+1}-{C}_i}{\varDelta x} $$
(14)

where D is the diffusion coefficient of the solute, C i and C i + 1 are the concentration of solute in segment i and i + 1, respectively, and ∆x is the segment width. The change in solute concentration ΔC i over a time interval ∆t of a minute at segment i is expressed by

$$ \frac{\varDelta t{C}_i}{\varDelta t}\frac{S_i+{S}_{i-1}}{2}\varDelta {x}_i={J}_i{S}_i-{J}_{i-1}{S}_{i-1} $$
(15)

where S i and S i − 1 are the sectional areas of segments i and i − 1, respectively, and S i is expressed as

$$ {S}_i=\frac{1}{2\sqrt{3}}{\left({\displaystyle \sum_{k=1}^i}\varDelta {x}_k\right)}^2 $$
(16)

From Eqs. (15) and (16), ΔC i can be found as

$$ \varDelta {C}_i=\frac{2D\varDelta t}{\varDelta {x}_i\left({S}_i+{S}_{i-1}\right)}\left\{{S}_i\frac{C_{i+1}^B-{C}_i^B}{\varDelta {x}_i}-{S}_{i-1}\frac{C_i^B-{C}_{i-1}^B}{\varDelta {x}_{i-1}}\right\} $$
(17)

where C B i + 1 , C B i and C B i − 1 are the solute concentrations in segment i + 1, i and i − 1 at the previous ∆t. Assuming the solute in a dendrite obeys the mass conservation law, we get

$$ \begin{array}{c}\hfill {\left({\displaystyle \sum_{k=1}^N}\kern0.5em \varDelta {x}_k\right)}^2{C}_0={\displaystyle \sum_{i=1}^N}\left\{\left({\displaystyle \sum \kern0.5em _{k=1}^i}\varDelta {x}_k+{\displaystyle \sum_{k=1}^{i-1}}\kern0.5em \varDelta {x}_k\right)\varDelta {x}_i{C}_i^S\right\}\hfill \\ {}\hfill +{\displaystyle \sum_{i=j+1}^N}\left\{\left({\displaystyle \sum_{k=1}^i}\kern0.5em \varDelta {x}_k+{\displaystyle \sum \kern0.5em _{k=1}^{i-1}}\varDelta {x}_k\right)\varDelta {x}_i{C}_i^L\right\}\hfill \end{array} $$
(18)

where N is the total number of segments, C 0 is the initial solute concentration, C S i is the solute concentration in the solid phase segment i, C L i is the solute concentration in the liquid phase (assuming the solute concentration in the liquid phase is uniform), and j is the segment number of the solid phase at the solid/liquid interface.

During the solidification process, the solute concentration at the solid/liquid interface is determined by the non-equilibrium concept as follows:

$$ {C}_j^S={k}_V{C}_{j+1}^L $$
(19)

where k V is the non-equilibrium distribution coefficient defined by Eq. (5), and C S j and C L j + 1 are the respective solute concentrations of the solid and liquid phases at the solid/liquid interface.

In the solidification segregation calculation mentioned above, solidification was determined by the solidification initiation temperature (T I ) and the calculated result of Eq. 13. This assumes that the solidification of the secondary phase (316FR-A: δ-ferrite, 316FR-B: austenite) occurs when the liquidus temperature of the residual liquid phase is reduced to below the dendrite tip temperature (T Tip ) of the secondary phase, which was also calculated from Eq. 13. The segregation computation is then subsequently progressed for each respective phase until solidification reaches completion. The liquidus temperature in the residual liquid phase during solidification segregation (T m ) was assumed from the decrease in liquidus temperature that corresponds to the segregated concentration from the initial solute composition, taking into account the velocity-dependent liquidus gradient as follows:

$$ {T}_m={T}_{Tip}+\kern0.5em {\displaystyle \sum {m}^V}\left({C}_{j+1}^L+{C}^0\right) $$
(20)

where T Tip is the dendrite tip temperature calculated from Eq. 13, and m V is the velocity dependent liquidus gradient from Eq. 4. The decrease in liquidus temperature was determined as from the sum of each binary system’s contribution (i.e. Fe-Cr, Ni, Mo, Si, Mn, P, S, N and C).

Solidification segregation was computed at 95 % solidification completion (i.e. the fraction of residual liquid phase was 5 %), which was based on a previously observed tin-quenched solidification microstructure [30–32]. The segregated concentration of all solute elements in the residual liquid phase at 95 % solidification completion was inputted into Thermo-Calc software (SSOL4 database), and the resulting equilibrium solidus temperature was adopted as the solidification completion temperature (T C ) in the same manner as in [30–32].

4.3 Conditions for numerical calculation

Tables 4 and 5 summarise the material constants used in the calculation [30–35]. The equilibrium liquidus temperature (T eq L ) of the steel was obtained using Thermo-Calc software with the SSOL4 database. For simplicity, the calculation was carried out under the following assumptions: Dendrite solidification velocity is equal to welding speed, cooling rate remains constant during solidification, and interaction between solute elements (i.e. cosegregation) is negligible. These calculation conditions are listed in Table 6. The dendrite radius employed was 10.0 μm, and the mesh was divided into 100 segments. Welding speeds of 1.67 (GTAW), 20.0 (LBW) and 40.0 mm/s (LBW) were used, and the cooling rate during solidification was assumed to be 240 K/s (1.67 mm/s), 1200 K/s (20.0 mm/s) and 1950 K/s (40.0 mm/s) based on the measured thermal cycle mentioned in Sect. 2.2.

Table 4 Material constants used for numerical calculation of solidification initiation temperature (for the KGT model)
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Table 5 Material constants used for numerical calculation of solidification initiation and completion temperatures in binary systems of Fe-C, S, P, Cr, Ni, Si, Mn, Mo and N (for KGT and solidification segregation models)
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Table 6 Conditions for numerical calculations
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4.4 Calculation results

4.4.1 Solidification segregation behaviour

Of the various alloying elements used in stainless steels, sulfur (S) and phosphorous (P) have generally been regarded as the most significant with regards to susceptibility to solidification cracking (especially when it comes to cracking temperature range), and segregation of the liquid phase remaining between dendrites during solidification [11–17, 25, 30–32]. Figures. 7 and 8 show the segregation behaviour of S and P during the solidification for 316FR-A (Fig. 7) and 316FR-B (Fig. 8). In the case of 316FR-A (Fig. 7), the solidification of austenite (indicated by arrows) progressed from the left and right ends of the δ-ferrite at each welding speed, with the opposite being true of 316FR-B (Fig. 8). The composition of the residual liquid phase at the completion of solidification (i.e. solidified fraction 95 %) was located between the respective peak concentrations of the δ-ferrite and austenite phases for each welding speed and steel type. As solidification progressed, the concentration of S increased in both the δ-ferrite and austenite phases, resulting in a large amount of S segregating into the remaining liquid phase after solidification was complete, regardless of welding speed. Increasing the welding speed did, however, makes elements more easily distributed in the solid phase. Figure 9 shows the change in the segregated concentration of the residual liquid phase as a function of welding speed at the completion of solidification for both 316FR-A and 316FR-B. Note that the segregated concentration of S increases with welding speed, with a similar tendency seen with the impurity element of P. Thus, it follows that the solidification segregation behaviour of impurity elements such as S and P during solidification after LBW is not mitigated by welding speed with either 316FR-A or 316FR-B.

Fig. 7
figure 7

Calculated solidification segregation history of a S and b P for 316FR-A (*δ: δ-ferrite, γ: austenite)

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Fig. 8
figure 8

Calculated solidification segregation history of a S and b P for 316FR-B (*δ: δ-ferrite, γ: austenite)

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Fig. 9
figure 9

Relationship between welding speed and segregated concentration of a S and b P in residual liquid phase at solidification completion

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4.4.2 Effect of welding speed on the temperature range of solid/liquid coexistence

Figure 10 shows the calculated results for the solidification initiation (T I ), the completion temperature (T C ) and the solid/liquid coexistence temperature range (T I  − T C ) as a function of welding speed for 316FR-A (Fig. 10a) and 316FR-B (Fig. 10b). It is evident from this that with both steels, T I decreases with an increase in welding speed due to the increased supercooling (predominantly constitutional undercooling behaviour). Although T I  − T C increases for both 316FR-A and 316FR-B, the magnitude of this increase is greater in the case of 316FR-B because of the greater decrease in T C . In other words, the difference in the variation of BTR with welding speed between 316FR-A and 316FR-B strongly correlates to the variation in the behaviour of T C , which is mainly dominated by the segregation of impurity elements (i.e. S and P). Figure 11 additionally shows the relationship between the experimentally determined value of BTR and T I  − T C , wherein we see that although the absolute value of T I  − T C deviated from the experimental BTR, the BTR obtained through transverse-Varestraint testing showed a positive correlation to T I  − T C . Consequently, the mechanism by which BTR varies could be regarded as being related to the variation mechanism of T I  − T C , especially considering the solidification segregation behaviour of the impurity elements S and P.

Fig. 10
figure 10

Calculated solidification initiation temperature, solidification completion temperature, and solid/liquid coexistence temperature range as a function of welding speed for a 316FR-A and b 316FR-B

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Fig. 11
figure 11

Relationship between BTR and solid/liquid coexistence temperature range

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4.5 Mechanism for variation in the solidification cracking susceptibility

As a general rule, solidification segregation is known to be mitigated during rapid solidification (such as that which follows LBW) [11, 36, 37] due to an increased partitioning coefficient (i.e. solute trap). Figure 7 also indicates that this solute trap phenomenon follows Eq. 5, but although solute trapping occurred in both phases during the solidification of 316FR-A and 316FR-B, the solidification segregation behaviour of the impurity elements was not alleviated, as already explained in Fig. 9.

The formation of δ-ferrite during solidification has typically been regarded as playing a positive role in reducing susceptibility to solidification cracking, as the ferrite phase has a higher solubility for impurity elements (such as S and P) than the austenite phase [12, 13]. The amount of δ-ferrite at the completion of solidification was calculated based on the model depicted in Fig. 6 (i.e. the area fraction of the δ-ferrite phase = the area of δ-ferrite phase / the total area of the half-quadrangle (ABC)). Figure 12 shows the calculated result for the amount of δ-ferrite at the completion of solidification as a function of the welding speed. It is evident from this that the ferrite content at solidification completion decreases with an increase in welding speed from 16 to 10 % in the case of 316FR-A, and from 50 to 30 % with 316FR-B. Although the ferrite content decreased with increasing welding speed with both materials, the decrease with 316FR-B (20 %) was greater than that for 316FR-A (6 %); i.e. the decrement tendency was different between the two. Thus, through consideration of the solidification segregation of impurity elements and the ferrite content during welding solidification, the reason for the greater increment in BTR with 316FR-B can be considered a result of the greater decrease in ferrite causing a more segregation of impurities. This, in turn, increases the temperature range of solid/liquid coexistence when compared with that of 316FR-A.

Fig. 12
figure 12

Relationship between welding speed and calculated amount of δ-ferrite at solidification completion

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5 Conclusions

The solidification cracking susceptibility of the two different solidification modes of 316FR stainless steel (316FR-A: AF mode solidification, 316FR-B: FA mode solidification) laser welds was quantitatively evaluated by newly developed LBW transverse-Varestraint test. Through this, it was shown that as the welding speed increased from 1.67 (GTAW) to 40.0 (LBW) mm/s, there was an increase in BTR with both 316FR-A (from 37 to 46 K) and 316FR-B (from 14 to 46 K) weld metals. However, as the 316FR-B had a greater range of BTR increase, it follows that the variation in solidification cracking susceptibility behaviour with welding speed (i.e. the application of LBW) varied in accordance with the solidification mode of the steel used. The reason for this greater increment of BTR with 316FR-B could be explained by the larger decrease in the amount of δ-ferrite during solidification, which affects the segregation of impurities (namely, S and P) and increased the temperature range of solid/liquid coexistence when compared with 316FR-A.