Hot cracking in stainless steel 310s, numerical study and experimental verification

Computational Materials Science 63 (2012) 182–190

Contents lists available at SciVerse ScienceDirect

Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci

Hot cracking in stainless steel 310s, numerical study and experimental verification A.R. Safari a,⇑, M.R. Forouzan a, M. Shamanian b a b

Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran Department of Materials Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran

a r t i c l e

i n f o

Article history: Received 29 February 2012 Received in revised form 8 June 2012 Accepted 11 June 2012 Available online 7 July 2012 Keywords: Hot cracking Austenitic stainless steel Viscosity Constitutive model Annealing

a b s t r a c t Hot cracking is a serious problem in welding of many alloys such as high strength steels, austenitic stainless steels and aluminum alloys. It takes places during the last stage of solidification where mechanical deformation develops in the mushy zone while the material has low ductility. In this study, hot cracking initiation and propagation of austenitic stainless steel 310s was studied. A viscoplastic constitutive model was proposed and implemented in finite element simulation. Solidification shrinkage, viscosity, annealing at high temperature and melting effect of fusion zone and a criterion for hot cracking initiation and propagation are the main features of the solution. Numerical results were compared with some experiments accomplished in this study in order to verify the proposed method. The results showed that maximum transverse mechanical strain criterion could predict both initiation and propagation of the hot cracking. Annealing and melting had the most effect in predicting crack length. Omission of these parameters leaded to underestimated results. Elimination of viscosity effect leaded to overestimation of the crack length while elimination of solidification shrinkage leaded to underestimated crack length. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction In the welding process, a concentrated moving heat is applied to the joint. This results in a non-uniform transient temperature field in the structure. Because of rapid cooling rate, the solidification of fusion zone is a non-equilibrium process with dendritic microstructure in the solidified weld metal. During the last stage of the solidification, low melting-point constituents segregate between solidifying dendrites and form liquid films over the interface of dendrites. This stage of solidification in which both liquid and solid coexist is called brittle temperature range (BTR). The BTR starts from a temperature below the liquidus temperature called coherency temperature (TC). Below the TC, dendrite arms coalesce; solid skeleton form and material can transmit stress. In the BTR, the ductility of material is low. At the same time, material contracts from a state of zero stress and zero strain and tensile deformation develops due to external restrains. Hot cracking occurs when tensile deformation exceeds from material strength and reflow of melt cannot heal them. Weld metal contracts due to both thermal contraction and solidification shrinkage. High thermal contraction of austenitic stainless steels makes them prone to solidification cracking. In aluminum alloys, both thermal contraction and solidification shrinkage are high. Therefore, some aluminum alloys especially those with wide BTR are prone to solidification cracking ⇑ Corresponding author. Tel.: +98 311 3915235; fax: +98 311 3912628. E-mail addresses: [email protected] (A.R. Safari), [email protected] (M.R. Forouzan), [email protected] (M. Shamanian). 0927-0256/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2012.06.015

[1]. Weld metal solidification cracking has the same root of cracking in casting. In fact, many ideas for weld metal solidification cracking come from studies about cracking in casting which is usually referred to as ‘‘hot tearing’’. More details about hot crack formation can be found in [2,3]. A summary of hot cracking mechanisms can be found in [4–6]. Various factors have effect on materials susceptibility to hot cracking. Low melting point constitutes increase hot cracking susceptibility. During solidification, this constitutes are rebutted by solidifying dendrites and concentrates at the weld centerline increasing crack susceptibility. Solidification temperature range, temperature gradient, solute content at the last stage of solidification, surface tension of interdendritic liquid, impurities, ductility of weld metal in mushy zone, thermal contraction, solidification shrinkage, extent of restrain, grain shape, structure and size, are among the most important parameters affecting materials susceptibility [1,5]. The metallurgical aspects of hot cracking have been studied in much more details than the thermo mechanical aspects. A successful description of hot cracking should take into account both metallurgical and mechanical aspects. Metallurgical aspects consider microscopic features and weldability tests consider these effects. Mechanical aspects consider macroscopic features and numerical methods such as finite element can be used to estimate them. Finite element simulation of weld solidification cracking returns to early years of 1990s. Feng made a considerable contribution to the precise modeling of hot cracking [7]. He studied a bead-on-plate weld of an aluminum 2024 plate and evaluated

A.R. Safari et al. / Computational Materials Science 63 (2012) 182–190

the mechanical strains near the weld pool. Yang et al. simulated a trailing heat sink as a hot-cracking mitigation technique [8]. They used finite element method and ductility curve in the BTR for simulation. Wei et al. studied hot cracking of stainless steel 310s [9]. They used finite element method to model mechanical strain behind the weld pool as driving force and ductility curve as material resistance. Ductility curve was extracted by method of in situ observation (MISO). Xu et al. proposed trailing impactive electromagnetic force to control hot cracking [10]. Olivier et al. evaluated the capability of various kinds of hot tearing criteria to predict the initiation of hot tears in casting [11]. According to their results, Prokhorov and Won et al. criteria agreed well with the experimental observations. Most studies for simulation of hot cracking are based on the prediction of crack occurrence but not on its propagation. Dike et al. used finite element method with interface element and damage model of Bammann to study weld solidification cracking of 6061-T6 aluminum [12]. Good agreement was obtained between simulation and experiment for location of crack initiation and extent of cracking. Shibahara et al. proposed a temperature-dependent interface element for simulating occurrence and propagation of hot cracking [13]. In this method, crack propagates when required energy for formation of new surface is provided. Hilbinger et al. used liquid element for accounting local tensile stress in middle of the weld seam [14]. These elements had very low yield strengths in the BTR range. They used a criterion based on the maximum deformation that the liquid elements in the solid–liquid region can sustain. This critical deformation was determined experimentally. In this study, hot cracking of austenitic stainless steel 310s was studied. The main contribution was to include not only crack initiation but also crack propagation. Maximum transverse mechanical strain was successfully used for the first time as the criterion for modeling crack propagation. Another important contribution was to study the combined effect of annealing at high temperature and melting of fusion zone on the crack propagation. The effect of welding parameters, solidification shrinkage and viscosity on hot cracking was also studied. Finite element method was used and a viscoplastic constitutive model with internal state variables and temperature dependent material properties was developed. Elastic and viscoplastic strains were also treated as state variable (a total number of 26 solution dependent state variable). Viscoplastic constitutive behavior of material was extracted by static and dynamic hot tensile tests. The material behavior of fusion zone was also included in the developed constitutive behavior. Numerical result was verified by some experiments accomplished in this study. 2. Hot cracking criteria 2.1. Prokhorov criterion Prokhorov proposed the first quantitative criterion for hot tearing [15]. Prokhorov criterion is based on ductility curve as material strength and deformation of the liquid–solid region as driving force. Using a rapid tensile type test, he extracted ductility curves of aluminum alloys in the BTR. Fig. 1 shows the ductility curve typically. Weldability tests with controlled strain such as Trans-Varestraint test (TVT) can also be used to extract ductility curve. According to this criterion, cracking occurs when strain rate of material in the BTR exceeds the critical strain rate for temperature drop (CST). The CST is obtained from ductility curve. Prokhorov hot cracking index (HCI) is defined according to the following equation:

 emin _  HCIprokhorov ¼ e_  jTj BTR

ð1Þ

183

Fig. 1. Schematic of ductility curve in the BTR.

Cracking occurs when HCIprokhorov P 0. Prokhorov criterion is purely mechanical and do not consider metallurgical aspects. 2.2. Clyne and davies criterion Clyne and Davies divided the mushy zone into liquid feeding zone (0.4 < fs < 0.9) and brittle temperature zone (0.9 < fs < 0.99) where fs is solid fraction [16]. Cracks formed in the liquid feeding zone are healed by surrounding melt, whereas cracks formed in the BTR zone cannot be healed, because the dendrite arms are close enough to resist feeding of the surrounding liquid. They introduced a criterion equal to tv/tr. tv is the time spent by the mushy zone in the feeding region and tr is the time of solidification in the BTR in which solid content increases from 0.9 to 0.99. Therefore, the Clyne’s hot cracking index is defined as:

HCIclyne ¼

t0:99  t0:9 P0 t0:9  t 0:4

ð2Þ

Indeed, this criterion is based on thermal criterion only and no mechanical aspect is considered. 2.3. Yamanaka criterion According to this criterion, Cracking occurs when accumulated inelastic strain of mushy zone exceeds from a critical value [17]. By means of tensile hot cracking test and MISO technique, Matsuda et al. obtained the critical strain for austenitic stainless steel 310s [18]. The results depended on the strain rate and were in the range of 0.01–0.019. Won et al. suggested a simple empirical relation for the critical strain of low-alloyed carbon steels [19]. It takes into account the strain rate and brittle temperature range.

u ecr ¼ _ m n e BTR

ð3Þ

u, m and n are material constant and can be found experimentally. According to Eq. (3), with increasing strain rate or increasing the BTR, the critical strain decreases and so the crack susceptibility increases which is in agreement with experimental observations. Again vulnerable region (0.9 < fs < 0.99) is considered as the hot cracking susceptible region and accumulated inelastic strain is accounted for this region. Hot cracking index is according to the following equation:

184

A.R. Safari et al. / Computational Materials Science 63 (2012) 182–190

HCIwon ¼

fs X ¼0:99

Dei:e:  ecr P 0

ð4Þ

fs ¼0:9

2.4. RDG criterion

in processes such as welding and casting that the behavior of material at high temperature is intended, a viscoplastic constitutive model should be used. Based on the excess stress theory, Perzyna proposed a viscoplastic constitutive equation in the form [24]:

1

Rappaz et al. proposed a criterion based on the pressure drop at the dendrite root [20]. Pressure drop is due to insufficient liquid feeding for compensating deformation and solidification shrinkage and results to the formation of a void. The void then develops into a crack. According to this criterion hot cracks initiate when the local pressure between the dendrite tips and roots drops below a given cavitation pressure. It accounts the deformation of the coalesced solid skeleton (mechanical deformation) and shrinkage perpendicular to dendritic arms. The maximum pressure drop can be obtained by the following equation:

Dpmax ¼ Dpme þ Dpsh 180 ð1 þ bÞl ¼ 2 G k 

fS ðTÞ2 ð1  fS ðTÞÞ2

Z

TL

TS

EðTÞfS ðTÞ2 ð1  fS ðTÞÞ3

dT þ

180

v T bl

k2

G

Z

ð9Þ

where W(F) is a function of the static yield function F. Varies functions for W(F) has been proposed. In a special case, W(F) = F and F ¼ f ðrij ; T; eijv p Þ  RðT; HÞ where R is yield strength and H is hardening parameter [25]. In this special case, viscoplastic strain rate is according to following equation:

_ vp

eij

+ rffiffiffi * 12 31 3 Sij  aij ¼ ðSij  aij ÞðSij  aij Þ  RðT; HÞ 2l 2 kSij  aij k

ð10Þ

aij are back stress tensor components. Wang and Inoue showed that TL

TS

dT

ð5Þ

R where EðTÞ ¼ G1 fS ðTÞe_ P ðTÞdT, b is shrinkage factor, l is liquid viscosity, k is secondary dendrite arm spacing, G is thermal gradient, vT is isotherms velocity, fS(T) is solid fraction at any temperature T, TL is liquidus temperature and TS is coherency temperature. Hot cracking susceptibility is defined as the inverse of the maximum strain rate sustainable by the mushy zone. The RDG hot cracking index is defined according to the following equation:

HCIRDG ¼ ðDpmax  Dpcr Þ P 0

ð6Þ

in this situation, constitutive relationship of Perzyna could describe the materials behavior over a wide range from inelastic solid to viscos fluid [25]. More general representation of Perzyna viscoplastic model is as follow:

e_ ijv p ¼ KhWðFÞim

Ploshikhin et al. presented an integrated mechanical–metallurgical approach to model solidification cracking in welds [21]. It undertakes the effects of strain accumulation at the final stage of solidification, microstructure of the mushy zone and thermodynamical properties of the welded material. According to Ploshikhin criterion, hot tearing takes place when deformation of the interdendritic liquid film exceeds from some critical value. Maximum deformation is a function of welded joint geometry, welding parameters, thermo physical properties of base and parent metal and parameters of the microstructure. Ploshikhin hot cracking index is defined according to the following equation:

  cr HCIPLOSHIKHIN ¼ dmax acc  dacc P 0

@F @ rij

ð11Þ

For fluids, the exponent m tends to unity and K = (3l)1. Other constitutive equations have also been proposed for steels at high temperature. Based on curve fitting of experimental results, Kozlowski et al. proposed four one-dimensional constitutive equations to model mechanical behavior of plain carbon steel in the austenite phase [26]. According to their results, model III had the best compatibility with experiment. Eq. (12) presents threedimensional extension of model III.

2.5. Ploshikhin criterion

e_ ijv p ¼

rffiffiffi 3 S   aan im ij Kh r 2 kSij k

ð12Þ

where



Q KðTÞ ¼ CðcÞ exp  T

a_ ¼ e_ v p CðcÞ ¼ 46550 þ 71400c þ 12000c2 Q ¼ 44650

ð7Þ a ¼ 130:5  5:128  103 T

Other hot cracking criteria can be found in [4,6,22,23].

n ¼ 0:6289 þ 1:114  103 T

3. Constitutive behavior Considering additive decomposition of strains, total strain rate could be decomposed into its components.

_ e _ v p _ th _ Trp _ Trv e_ Tot ij ¼ eij þ eij þ eij þ eij þ eij

@F

e_ ijv p ¼ hWðFÞi @ rij l

ð8Þ

The terms on the right hand side of the above equation are elastic, viscoplastic, thermal, transformation plasticity and volumetric strain rate due to phase transformations respectively. At relatively low temperatures, an elastic–plastic constitutive relationship can describe the material behavior well. Power relationships are usually used for this purpose. However, with increasing the temperature, viscosity effect increases so that beyond the melting point, the material behaves like a viscous liquid. Therefore,

m ¼ 8:132  1:54  103 T Q is activation energy constant and c is carbon content. Comparison of varies constitutive laws showed that most of them have forms identical to the following equation: 1

r ¼ RðT; ev p Þ þ LðTÞe_ v p =mðTÞ

ð13Þ

n where R ¼ ry ðTÞ þ KðTÞev p ðTÞ is static yield surface. For example,

compare to Kozlowski model III:

RðT; ev p Þ ¼ aðTÞev p

nðTÞ

and LðTÞ ¼

1 Q expð Þ CðcÞ T

1=mðTÞ

A.R. Safari et al. / Computational Materials Science 63 (2012) 182–190

At low temperature or low strain rate, the second term of Eq. (13) vanishes and the model reduces to a simple power law. For treatment of mushy zone, two rheological behaviors was proposed by Koric and Thomas [27]. In the first approach, an elastic-perfectly plastic rate-independent constitutive equation with small yield strength was assigned to the mushy zone. The assigned yield strength was in the order of 0.01 Mpa to enforce negligible stresses in the mushy zone. In the second approach, a viscoplastic constitutive model according to Eq. (14) was proposed.

c e_ v p ¼ s hjcs r  j  ry i

ð14Þ

lv

Where

( emax j

cs ¼

emin jemax

emin emax P emin : emax < emin

8 l1 v is a large enough coefficient in the order of 10 . In this study, the first approach was adopted for mushy zone.

4. Integration algorithm In this study, a temperature and strain rate dependent kinematic hardening model incorporating viscosity effects was developed and incorporated as the constitutive behavior of material. Stress–strain relaxation of fusion zone was also included. To carry out relaxation, elastic and plastic strains were set as state variable. A total number of 26 solution dependent state variables were used. The radial return method was employed to calculate the stress components. According to this method, for each increment n + 1: nþ1

sij ¼ nþ1 strij  3nþ1 Gn gij Dep

ð15Þ

where n

gij ¼

nþ1

nþ1 tr sij

 nþ1 bn aij r tr

b ¼ nþ1 H=n H

G is the shear modulus. For bilinear kinematic hardening model, the evolution of back stress is according to Eq. (16) from which back stress at the end of each increment can be find.

d aij  2 p ¼ e_ ij dt H 3 nþ1

ð16Þ

aij ¼ nþ1 bn aij þ nþ1 HDepn gij

ð17Þ

From Eqs. (15) and (17): nþ1  tr

r  nþ1 r  ð3nþ1 G þ nþ1 HÞDep ¼ 0

ð18Þ

In addition, from Eq. (13) for kinematic hardening:

Dev p ¼

nþ1

r  ry ðnþ1 TÞ

5. Annealing and melting effects of the heat source In thermo mechanical processes, when the material temperature exceeds from 0.6 to 0.7 of the melting temperature (Tm), annealing starts [28]. In annealing, previous history of work hardening clears. This means that annealing resets the equivalent plastic strain and any hardening [29]. On the other hand, during the movement of the welding arc, boundary of the fusion zone continuously changes and so the elements along the fusion line enter and exit from the weld pool in a dynamic nature. When elements enter the weld pool, their previous strain and stress is relaxed. Behind the fusion zone, material starts to solidify and stresses and strains accumulate from a state of zero. In this study, for assessing fusion zone, strain relaxation method is implemented. In this method, before mechanical analysis, temperature history of all elements from thermal analysis is modified and temperatures above Tm are truncated to Tm. Melting temperature is also defined as the thermal expansion reference temperature. Therefore, elements will be expansion-free until the temperature of nodes is cooled down below Tm. This method has the same effect of element birth and death method for modeling addition of weld metal. Modification of temperature history was accomplished by a specially developed subroutine. For structural analysis, in the developed constitutive subroutine, annealing was treated as a linear function of temperature, starts from 0.6 Tm and completes at Tm. For incorporating the melting, when the temperature reaches to Tm, elastic–plastic strains and stresses are set to zero. 6. Model geometry and material properties Many weldability tests have been developed to quantify hot cracking susceptibility. Some tests such as varestraint use an externally applied load (predefined stress or strain) as the driving force and some others such as houldcroft test are self-restraint. Maximum crack length or summation of all crack lengths is commonly used as a measure of crack susceptibility. In this study, three nominally identical specimens were manufactured from a 3 mm thick plate of AISI Type 310s steel. Test specimens were cut similar to the houldcroft test model. Dimensions of the test specimens are illustrated in Fig. 2. To eliminate fabrication residual stresses, specimens were solution heat treated at 1050 °C for 45 min and then cooled in water. For each specimen a single autogenously fusion line was deposited along the centerline. The arc started from tab plate. When fusion reaches to the bridge, tab plate falls and a steady fusion continues on the test specimen. For each specimen two K-type thermocouples were used to record the temperature. To minimize the effect of shielding gas on the records, thermocouples were installed on the back of plate with 10 mm distance from weld centerline. Thermocouples were inserted in holes with 1 mm diameter and 1 mm depth. Pure nitrogen gas was used as the shielding gas. For protecting the fusion

mðnþ1 TÞ

Lðnþ1 TÞ

Dt

ð19Þ

Combination of Eqs. (18) and (19) leads to the following equa. tion that should be solved to find nþ1 r nþ1 

r  nþ1 r  ð3nþ1 G þ nþ1 HÞ

nþ1

r  ry ðnþ1 TÞ Lðnþ1 TÞ

mðnþ1 TÞ

Dt ¼ 0

ð20Þ

 ; Dev p can be find from Eq. (19). Obtaining nþ1 r In this study, the Von Misses yield surface and its associated flow rule with bilinear kinematic hardening model was used. Large deformation was accounted in all analyses. Various consistent tangent modules were examined to increase the rate of convergence.

185

Fig. 2. Dimensions of the test specimen (mm).

186

A.R. Safari et al. / Computational Materials Science 63 (2012) 182–190

temperature history of nodes. Following the thermal analysis, a sequentially coupled thermal-stress analysis was performed 8. Thermal analysis The Goldak double ellipsoid volumetric heat flux and Gaussian surface heat flux was examined. Both models had approximately identical results but surface heat flux was more compatible with the experiments. It seems to be due to omission of consumable electrode and small thickness of specimens. The model uses following equation to describe the heat source:

qðx; y; tÞ ¼

Fig. 3. Test setup.

zone from oxidation, a box were constructed and installed on the back of the plate to blow nitrogen. Welding was performed at the ambient temperature of 10 °C. Type of welding was automatic tungsten inert gas welding (TIG). Heat input was 585 J/mm, welding speed was 1 mm/s and arc time was 80 s. Total weld length was 80 mm. Fig. 3 shows the test setup. Parameters of Eq. (1) were obtained by static and dynamic tensile tests at low and high temperatures. A serious of tension tests were accomplished at a velocity of 1.5 mm/min (strain rate 0.001 s1) and temperatures of 22 °C, 250 °C, 500 °C, 700 °C, 900 °C and 1000 °C. Other tension tests were accomplished at a velocity of 225 mm/min (strain rate 0.15 s1) and temperatures of 700 °C, 900 °C and 1000 °C. 7. Finite element analysis In this study, a 3D finite element model was used. Because of the symmetry about the weld centerline, only half of the specimen was modeled. Fixed boundary condition was used for the grip area. The specimen was meshed so that at the position of each thermocouple one node was located. Fig. 4 shows the finite element mesh. Finer mesh was used near the weld centerline (WCL). Elements away from the WCL had larger size to decrease solution time. Thermal stress analysis was performed in two distinct steps using the commercial software ABAQUS [29]. Since the temperature field is independent of stress or displacement solution, transient heat transfer was performed first to determine the

Fig. 4. Mesh density for thermal and stress analysis.

( ) 3Q 3½ðx  ðx0 þ v ðt  t 0 ÞÞÞ2 þ y2  exp r20 pr20

ð21Þ

where Q = gVI is the power input. g is the arc efficiency, V is the voltage and I is the current. r0 is a characteristic length and is such that the heat flux at the boundaries falls to 5% of its maximum value. v is the arc speed, t is the time and t0 is a lag factor denoting the dwell of torch at the start position. The user subroutine DFLUX in ABAQUS was used to introduce the heat flux. g and r0 were selected such that temperature prediction at the thermocouples position and predicted fusion dimension comply with experimental results. Dwell time was selected such that a stable fusion width was obtained. Thermal conductivity was increased by a factor of two in liquidus–solidus temp range to account stirring effect and fluid flow in the weld pool. Thermal boundary conditions include both convection and radiation heat lost through all sides of the plate exclude symmetry plane. Near the fusion zone, the effect of shielding gas increases the convection but dominant effect of emissivity compensates this. Convection coefficient and emissivity coefficient are assumed 15 (W/m2/°C) and 0.9 respectively. Initial temperature of specimen and ambient temperature were assumed 10 °C. The effect of phase change in the thermal analysis was taken into account by means of the latent heat of fusion. Release rate of the latent heat was assumed to be in direct proportion to the solid fraction. Fig. 5 shows the solidification fraction of sus310 as a function of temperature. According to Eq. (22), Latent heat can be incorporated in the specific heat as an equivalent specific heat.

Q ¼ qC p;eq

@T @t

where

Fig. 5. Variation of solid fraction with respect to temperature [18].

ð22Þ

A.R. Safari et al. / Computational Materials Science 63 (2012) 182–190

187

as an additional thermal expansion and was modified in expansion coefficient. The solidification shrinkage was neglected above the coherent temperature where the material is treated as a liquid with relaxed stress/strain. Fig. 7 shows modified thermal expansion coefficient. For elements that enter the fusion zone, the yield strength was assumed 0.01 Mpa and elastic/plastic strains were set to zero. In this study for elements that satisfy hot cracking index, yield strength of 0.01 Mpa was assigned. This causes that this elements treat as a very soft material after cracking. Won et al. criterion was used for prediction of cracking [19]. For initiation and propagation of hot cracking, critical strain was assumed 0.01. This is in the range that was obtained by Matsuda et al. by tensile hot cracking test [18]. For mechanical analysis, continuum three dimensional 8-node brick elements (C3D8) were used. Fig. 6. Equivalent specific heat.

10. Results and discussion

C p;eq ¼ C p 

L @fs

q @T

L is latent heat per unit volume. This leads to very nonlinear specific heat increasing solution time and convergence problems. Fig. 6 shows the equivalent specific heat. As a more convenient way, latent heat can be discretized in temperature steps proportional to the content of the liquid fraction. 8-node linear heat transfer brick elements (DC3D8) were used in thermal analysis. Fixed time stepping was used for the solution.

Fig. 8 shows test specimen 1 and its fusion shape. Experimental fusion width was about 6.05 mm on the top of the specimen and 5.24 mm on the back of that (after subtracting crack width). According to Fig. 9, Numerical FZ dimensions were 6 mm on the top of specimen and about 5 mm on the back of that. The results are in close agreement with experimental results. Fig. 10 shows measured temperature at thermocouple positions and simulated ones. It can be observed that the temperature values obtained with the numerical model coincide well with those measured experimentally. It seems that more accurate modeling of

9. Mechanical analysis The FE mesh used in the mechanical analysis was the same as that used in the thermal analysis. Mesh compatibility was accounted to import nodal temperatures from thermal result file. The temperature history from thermal analysis was corrected by an especially developed subroutine to be used by successive structural analysis. In the structural analysis, the developed constitutive subroutine was used to represent material constitutive behavior. During solidification, due to liquid–solid phase transformation and density difference between solid and liquid phase, the material contracts in excess of thermal contraction which is called solidification shrinkage. For austenitic stainless steel with FCC structure, the volumetric solidification shrinkage is about 4%, which is equivalent to a linear contraction of 1.3% [30]. In this study, solidification shrinkage was defined as a function of temperature in solidification temperature range and proportional to solid fraction. It was treated

Fig. 7. Equivalent thermal expansion coefficient.

Fig. 8. Fusion zone of test specimen (dimensions in mm).

188

A.R. Safari et al. / Computational Materials Science 63 (2012) 182–190

Fig. 9. Dimensions of fusion zone from numerical result (dimensions in mm).

Fig. 11. Experimental crack length of test specimen 1.

Table 1 Comparison of experimental and numerical crack length.

Experiment

Simulation

Specimen

Crack length (mm)

1 2 3 –

33 35 32 32

Fig. 12. Simulated crack area.

Fig. 10. Comparison of simulated temperature with experimental results.

heat loss from surfaces especially convection effect of torch shielding gas and back plate shielding gas could improve the results. In simulation of the welding, fusion width was more affected by g and r0 while temperature at thermocouples was affected by g, r0 and heat loss from free surfaces. Fig. 11 shows crack length of test specimen 1. To insure repeatability of results, three specimens were constructed and tested and Table 1 shows the crack length of them. Good agreement of results denotes that tests have been done under similar and controlled condition. Fig. 12 shows the specimen after cracking. Dark area shows elements that their center satisfies hot cracking index. The predicted crack length was about 32 mm that had good compatibility with experimental results. Fig. 13 shows maximum transverse mechanical strain (TEmax) along the WCL. According to this figure, at a

distance equal to 32 mm, TEmax is very low therefore, this curve can be used as an index to distinguish crack length. Large strain at the beginning of the diagram is due to large deformation of the cracked elements. Elimination of latent heat effect leads to a crack length of 31 mm (only 3% error) but welding parameters that influence on the weld pool geometry has significant effect on the crack length. An increase of 30% in heat input leads to a crack length of 52 mm (over 60% difference) while doubling the welding velocity from 1 mm/s to 2 mm/s leads to a crack length of 12.6 mm. These results show that hot cracking is very sensitive to heat input. The Prokhorov criterion can describe this behavior. With increasing the heat input, temperature gradient behind the weld pool decreases and lower strain rate is required to satisfy this criterion while increasing weld velocity has reverse effect. Fig. 14 shows the effect of various modeling parameters on predicted crack length. According to this figure, elimination of annealing, melting and solidification shrinkage, leads to lower crack length prediction. Annealing and melting has similar effect and elimination of them leads to the most underestimating crack

A.R. Safari et al. / Computational Materials Science 63 (2012) 182–190

Fig. 13. Maximum transverse mechanical strain along the WCL.

189

description of hot cracking should take into account both metallurgical and mechanical aspects. For indexing hot cracking various theories have been proposed most of them predict the occurrence of hot cracking but cannot predict crack propagation. In this study, theory of a maximum transverse mechanical strain was examined as a successful criterion for predicting occurrence and propagation of hot cracking of austenitic stainless steel 310s. Finite element method was used for analysis and extracting driving force. A thermo viscoplastic constitutive model with 26 solution dependent state variables and temperature dependent material properties was developed. Parameters of constitutive model were extracted by static and dynamic hot tensile tests. For thermal analysis, latent heat of fusion was introduced as a function of solid fraction that again reveals metallurgical aspects. In structural analysis, thermal expansion coefficient was modified taking into account liquid–solid transformation. Simulation results were verified by experimental tests. Three specimens was constructed and the method of welding was autogenous TIG welding, a method which is common in hot cracking susceptibility tests. According to this study solidification shrinkage, melting effect of fusion zone, annealing at high temperature and viscosity had significant effect for prediction of hot cracking. Omission of the first three parameters leaded to underestimating crack length while omission of the viscosity leaded to overestimated results. References

Fig. 14. Effect of various modeling parameters on predicted crack length.

length. Elimination of viscosity overestimates crack length. The rheological behavior of classical viscoplastic solids can explain the above behaviors. Consideration of solidification shrinkage leads to more thermal strain behind the weld pool. Because of external restrain, this leads to more driving force and larger mechanical strain will develop that leads to longer crack length. At constant thermal strain and thus driving force, consideration of viscosity leads to less mechanical strain because the rate dependent part of viscose behavior specify some of driving force to itself. With consideration of annealing and melting, the material behind the weld pool contract from a state of zero strain and tensile deformation will develop in material because While elimination of these parameters lead to compressive strains behind the weld pool. In the next step, the effect of critical strain on crack length was studied. According to the results, the more the critical strain, the smaller crack length will develop. Specification of a fictitious large critical strain, demands from crack initiation and propagation. According to Sections 2.1 and 2.2, the susceptible solid fraction range for hot cracking is from fS = 0.9 to fS = 0.99. To account this region in the analysis, the equivalent temperature range was extracted from the solid fraction curve (Fig. 5) and was implemented in the constitutive behavior. Elimination of this behavior and consideration of the full solid–liquid temperature range as the hot cracking susceptible temperature range (fS = 0 to fS = 1) resulted to a crack length of 34 mm. 11. Conclusion The metallurgical aspects of hot cracking have been studied in much more details than thermo mechanical aspects. A successful

[1] S. Kou, Welding Metallurgy, second ed., John Wiley & Sons, Hoboken, New Jersey, 2003. [2] H. Fredriksson, M. Haddad-Sabzevar, K. Hansson, J. Kron, Materials Science and Technology 21 (5) (2005) 521–529. [3] S. Kou, Journal of Minerals, Metals and Materials 55 (6) (2003) 37–42. [4] D.G. Eskin, Physical Metallurgy of Direct Chill Casting of Aluminum Alloys (Advances in Metallic Alloys), first ed., CRC Press, Taylor & Francis Group, 2008. [5] C.E. Cross, On the origin of weld solidification cracking, Hot Cracking Phenomena in Welds I, Springer-Verlag, Berlin Heidelberg, 2005. [6] Y.M. Wong, H.N. Han, T. Yeo, ISIJ International 40 (2) (2000) 129–136. [7] Z. Feng, A Methodology for Quantifying the Thermal and Mechanical Conditions for Weld Solidification Cracking, Ph.D. Dissertation, Ohio State University, 1993. [8] Y.P. Yang, P. Dong, J. Zhang, X. Tian, Welding Journal 79 (1) (2000) 9–17. [9] Y.H. Wei, Z.B. Dong, R.P. Liu, Z.J. Dong, Modeling and Simulation in Materials Science and Engineering. 13 (2005) 437–454. [10] W. Xu, H.Y. Fang, J.G. Yang, X.S. Liu, W.L. Xu, Materials Science and Engineering A 488 (2008) 39–44. [11] C. Olivier, C. Yvan, B. Michel, Journal of Engineering Materials and Technology 130 (2008) 021018-1. [12] J.J. Dike, J.A. Brooks, D.J. Bammann, M. Li, Thermal–mechanical modeling and experimental validation of weld solidification cracking in 6061-T6 aluminum, in: ASM International European Conference on Welding and Joining Science and Technology, Madrid, Spain, March 10–12, 1997. [13] M. Shibahara, H. Serizawa, H. Murakawa, Transactions of JWRI 28 (1) (1999) 47–53. [14] R.M. Hilbinger, H.W. Bergmann, W.Köhler, F. Palm, Consideration of Dynamic Mechanical Boundary Conditions in the Characterization of a Hot Cracking Test by Means of Numerical Simulation. Mathematical Modeling of Weld Phenomena 5, IOM Communications Ltd., London, pp. 847–862. [15] N.N. Prokhorov, Welding Production 4 (1962) 1–8. [16] T.W. Clyne, G.J. Davies, British Foundation 74 (1981) 65. See also: Brit. Found. 68 (1975) 238. [17] A. Yamanaka, K. Nakajima, K. Yasumoto, H. Kawashima, K. Nakai, Measurement of critical strain for solidification cracking, in: M. Rappaz, M.R. Ozgu, K.W. Mahin (Eds.), Modelling of Casting, Welding and Advanced Solidification Processes, vol. V, Davos, Switzerland, TMS, Warrendale, PA, 1990, pp. 279–284. [18] F. Matsuda, H. Nakagawa, S. Tomita, Transactions of JWRI 15 (2) (1986) 125– 133. [19] Y.M. Won, T.J. Yeo, D.J. Seol, K.H. Oh, Metallurgical and Materials Transactions B 31B (2000) 779–794. [20] M. Rappaz, J.M. Drezet, M. Gremaud, Metallurgical and Materials Transactions A 30A (1999) 449–455. [21] V. Ploshikhin, A. Prikhodovsky, M. Makhutin, A. Ilin, H.-W. Zoch, Integrated Mechanical–Metallurgical Approach to Modelling of Solidification Cracking in Welds, Hot Cracking Phenomena in Welds I, Springer-Verlag, Berlin Heidelberg, 2005. [22] D.G. Eskin, Suyitno, L. Katgerman, Mechanical properties in the semi-solid state and hot tearing of aluminium alloys, Progress in Materials Science 49 (2004) 629–711.

190

A.R. Safari et al. / Computational Materials Science 63 (2012) 182–190

[23] L. Katgerman, D.G. Eskin, In Search of the Prediction of Hot Cracking in Aluminium Alloys, Hot Cracking Phenomena in Welds II, Springer-Verlag, Berlin Heidelberg, 2008. [24] P. Perzyna, Thermodynamic Theory of Viscoplasticity, Advance of Applied Mechanics, vol. 11, Academic Press, New York, 1971. [25] Z.G. Wang, T. Inoue, Materials Science and Technology 1 (1985) 899–903. [26] P.F. Kozlowski, B.G. Thomas, J.A. Azzi, H. Wang, Metallurgical and Materials Transactions A 23 (1992) 903–918.

[27] S. Koric, B.G. Thomas, International Journal for Numerical Methods in Engineering 66 (12) (2006) 1955–1989. [28] M.C. Smith, A.C. Smith, International Journal of Pressure Vessels and Piping 86 (1) (2009) 79–95. [29] ABAQUS/standard user’s manual. Version 6.9.1. Providence. RI: ABAQUS Inc., 2009. [30] Doru Michael Stefanescu, Science and Engineering of Casting Solidification, second ed., Springer, New York, 2009.