Modelling of stress field during Submerged Arc Weld surfacing taking into account heat of the weld and phase transformation effect

In work the model of stress calculation and analysis of stress field during single-pass SAW (Submerged Arc Welding) surfacing have been presented. In temperature field solution the temperature changes caused by the heat of weld and by electric arc have been taken into consideration. Kinetics of phase changes during heating is limited by temperature values at the beginning and at the end of austenitic transformation, while progress of phase transformations during cooling has been determined on the basis of time-temperature-transformation (TTT) welding diagram. The stress state of thermal loaded flat has bean described assuming planar section hypothesis and simple Hooke’s law and using integral equations of stress equilibrium. Dependence of stresses on strains is assumed on the basis of tensile curves of particular structures, taking into account the influence of temperature. The analysis of stress state has been presented for SAW surfacing S355 steel plate.


Introduction
Modelling of thermomechanical states in the surfacing or rebuilding by welding, requires the determination of temperature field.It is necessary to calculate the shares of structural elements taking into account their changes that occur as a result of phase transformations.Finally the thermal and structural strains enable the determination of temporary and residual stresses.

Temperature field
In modelling of temperature field during welding dominate two approaches: analytical [1 -7] and numerical [8 -16], looking for a solution for particular welding methods and types of joints.In analytical description of the temperature

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where: T a (x,y,z,t) and T w (x,y,z,t) are temperature fields caused respectively by the heat of direct impact of an electric arc and by the heat of the weld reinforcement (consumed to melt the electrode), T 0initial temperature of material.
Then the geometry of the weld is presented in Fig. 1, where h w is the height of the weld (weld reinforcementthe part of the weld above the surface of the surfaced material), d p is the depth of material thickness loss (e.g.depth of wear zone), and l results from considering the volume of material supplied from the electrode.Analytical description of the temperature field caused by the direct impact of the electric arc with Gaussian heat distribution (Fig. 2) is shown in [18], whereas considering the heat stored in the liquid metal imposed on the surface is presented in [19].For accepted scheme of the single-pass surfacing (Fig. 3), the temperature field (1) is defined as follows: -for time t  t c , where t c is the total time of making of the weld: x 0 [m] and l [m] m are the coordinates of the start and length of the weld respectively.Quantity t 0 / s characterizes surface heat source distribution whereas r B 2 = 4at 0 [20] (compare Figure 2).Functions G 1 -G 4 and F 1 -F 4 are solutions of integrals using Gauss-Legendre quadrature [19].
Power of volumetric heat source accumulated in the molten electrode material is expressed by formula [21]: where: d is the diameter of the electrode, ρ e is the density of the electrode material and v e is wire feed speed, L is the heat of solidification [J/ kg], T e -the initial value of the outputted from the welding head electrode, T Lthe temperature at which the molten metal drop separates from the electrode.

Phase transformation kinetics
Kinetics of phase transformations during heating is limited by temperature values at the beginning (A 1 ) and at the end (A 3 ) of austenitic transformation.The amount of austenite  A created during heating the ferrite-pearlitic steel is defined according to the Johnson-Mehl-Avrami's and Kolomogorov's (JMAK) rule [22]: where  j 0 constitutes initial share of ferrite (jF), pearlite (jP) and bainite (jB), while constants b j and n j are determined using conditions of the beginning and the end of transformation: In welding processes the volume fractions of particular phases during cooling depend on the temperature, cooling rate, and the share of austenite (in the zone of incomplete conversion 0 A 1).In quantitative perspective the progress of phase transformation during cooling is estimated using additivity rule by voluminal fraction  j of created phase what can be expressed analogically to Avrami's formula by equation [23]: where:  j 0 is volume participation of j-th structural component, which has not been converted during the austenitization, T j s = T j s (v 8/5 ) and ) are respectively initial and final temperature of phase transformation of this component, , φ j max is the maximum volumetric fraction of phase j for the determined cooling rate estimated on the basis of the continuous cooling diagram (Fig. 5).while the integral volumetric fraction equals: The quantitative description of dependence of material's structure and quality on temperature and transformation time of over-cooled austenite during surfacing is made using the time-temperature-transformation diagram during continuous cooling, which binds the time of cooling t 8/5 (time when material stays within the range of temperature between 500 0 C and 800 0 C, or the velocity of cooling (v 8/5 = (800-500)/t 8/5 ) and the temperature with the progress of phase transformation (Fig. 5).Those diagrams are called TTTwelding diagrams.
The fraction of martensite formed below the temperature M s is calculated using the Koistinen-Marburger formula [24]: where  M denotes volumetric fraction of martensite, M s and

Thermal and structural strains
Total strain during single-pass surfacing represents the sum of thermal and structural strains during heating ( H ) and cooling ( C ): Then strains during heating is equal to: where:  iAstructural strain of i-th structure in austenite, T 0 initial temperature,  i -linear thermal expansion coefficient of i-th structure, and H(x) is the function defined as follows: During cooling, the strain can be described by the relation: where T SOL denotes solidus temperature, T sinitial temperature of phase transformation, T si -initial temperature of austenite transformation in i-th structure, γ Ai structural strain of austenite in i-th structure.In addition, due to the limit on solid state of material:

Model of stress calculation
Considered is longitudinally surfaced element, length of which is much bigger than its crosswise size.To describe stress state has been used prismatic rod subjected to mechanical strains, which for separate cross sections x are characterised by internal forces N = N(x) and M y = M y (x).Remaining forces are assumed to be negligible (transverse forces: T y = T y (x), T z = T z (x)) and absent (M x = M x (x)).The rod is also subjected to symmetric action in relation to z axis in slowly changing temperature field T = T(x,y,z) = T(x,-y,z).This field is characterised by low temperature gradient in relation to the variable x.The stress state of the flat is characterised by single dimensional stress state  x =  x (x,z,t) =  x (x,-y,z,t) (Fig. 6).

Figure 6. Scheme of rod subjected to mechanical and thermal loads
In order to derive formulas on strain and displacement of flat, Cauchy's relation has been used and planar section hypothesis has been assumed.In considered case, i.e. for technical theory of bent rods, differential equations of equilibrium are not used.Instead, integral conditions of equilibrium using simple Hooke's law  x = E x are applied [25].For modulus of longitudinal elasticity changeable towards coordinates (heterogeneous material of flat) or Young`s modulus dependent on temperature, stress can be described by: (31) where: The stresses in elasto-plastic state are determined by iteration using method of elastic solutions at the variable modulus of longitudinal elasticity conditioned by the stressstrain curve [26].Dependence of stresses on strains is assumed on the basis of tensile curves of particular structures, taking into account the influence of temperature.

Example of calculation
Calculations of the temporary temperature field for a square steel element with the side length 0,2 m and thickness of the plate 0,03 m made from steels S355J2G3 have been conducted.Thermal properties of welded subject material and electrode have been determined by a = 8•10 -6 m 2 /s, C p = 670 J/kg K,  = 7 800 kg/m 3 and L = 268 kJ/kg.
Numerical simulation has been conducted for the welding heat source of power 3 500 W, which corresponds to welding parameters (U = 30 V, I = 400 A,  = 0,95).In calculations there were assumed welding velocity v = 0,007 m/s, electrode wire diameter d = 3,5 mm, wire feed speed v e = 0,031 m/s and bead dimensions h w = 2,5 mm and w w = 22 mm (d p = 0).The initial value of temperature of electrode T e = 100 °C (a temperature of contact tip with the welding head).Computations have been made for middle crosssection of the surfaced element.
In Figure 7 maximum temperature distribution in cross section has been presented.The calculated isotherm 1 493 °C determines the fusion line and isotherms A 3 and A 1 determine the partial and full austenitization zones (Fig. 8).In the Figure 3, the selected cross section points were marked, for which an stress analysis were performed.The temperatures A 3 = 920 °C and A 1 = 748 °C have been calculated taking into account the effect of heating rate on these temperatures [27].The photograph of a metallographic macrosection in the middle cross section is shown in Figure 9. Calculated solidus isotherm 1 493 °C (solidification temperature of steel) -black line -corresponds to the fusion line obtained in the experiment.Bright line corresponds to the calculated temperature limit of the full transformation of austenite A 3 .
The phase transformations kinetics during heating is limited by the temperatures A 1 of the beginning and A 3 the end of the austenite transformation, while the progress of phase transformations during cooling was determined on the basis of TTT-welding diagram for S355 steel shown in Fig. 10 [28].).The heat cycles and changes in phase shares at selected points of the cross section (comp.Fig. 8) are presented in Figs.11 -14.In all of the thermal cycles peaks illustrate maximum temperatures during weld beads.Modelling of stress field during Submerged Arc Weld surfacing taking into account heat of the weld ….In strain calculations there were assumed linear expansion coefficients of particular structural elements and structural stresses (Tab. 1) determined on the basis of the author's own dilatometric research [29].Tensile curves of ferrite and pearlite are assumed on the basis of works [30,31].In case of austenite, bainite and martensite on the basis of data from works [32,33] tensile curves are determined according to Swift law [34].Tensile curve models of particular structures depending on temperature are presented in Figs. 15 -19.Residual normal stresses distribution in the middle cross section at the distance between -0.02 m and 0.02 m from the weld axis has been presented in Fig. 20, while for the whole cross-section of the element has been presented in Fig. 21.Similar distributions of normal residual stresses were obtained in experimental studies conducted by Chang et al. [35] and with use of FEM by Jiang et al. [36].In Figures 22 -25 the history of stress states changes at selected points of cross section has been presented (see.Fig. 8).At point 1 from the weld area (Fig. 22), stresses amount to zero as long as the point is in a state of liquid metal.After solidification during cooling tensile stresses increase.Sudden transformation of stresses into compressive is caused by phase transformation.At point 2 (Fig. 23) from the area of full transformation due to the heating, the compressive stresses occur.Then, due to transformation of initial structure into austenite, stress state changes into tensile.Next, tensile stresses, being under the influence of further temperature growth, decrease but during cooling stresses increase again.During phase transformations of overcooled austenite into ferrite, pearlite and bainite stresses plunge becoming compressive.Then during further cooling compressive stresses decrease but values of tensile stresses increase.At point 3 (Fig. 24

Conclusion
Presented model allows analysing and interpreting the influence of temperature field and phase transformations on strains and stresses caused by welding process using SAW method.Calculated temperature field, volume fractions of phases and stresses in longitudinally surfaced elements enable: (i) determination of heat affected zone, including areas of full and partial phase transformation as well fusion zone, (ii) analysis of stresses in any given point of section, investigation of phase volume fraction changes and strains and stresses caused by temperature changes and by phase transformations, (iii) analysis of stress states and plastic strain fields in rods during surfacing and residual stresses distribution after SAW surfacing.Calculated state of residual stresses is characterised by high values of tensile stresses in weld, fusion and full transformation zones as well as by sudden decrease of their values in partial transformation zone.Correctness of such residual stress distribution was proved experimentally and in numerical simulations using FEM by other authors.

Figure 1 .
Figure 1.Geometry of the weld

Figure 2 .
Figure 2. Gaussian distribution of heat source

Figure 5 .
Figure 5. Scheme of phase changes of overcooled austenite depending on cooling velocity within temperature range 800-500 0 C M f denote initial and final temperature of martensite transformation respectively, T the current temperature of process

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Endorsed Transactions on Industrial Networks and Intelligent Systems 01 2018 -03 2018 | Volume 4 | Issue 13 | e2Modelling of stress field during Submerged Arc Weld surfacing taking into account heat of the weld ….

Figure 19 .Figure 20 .Figure 21 .
Figure 19.Tensile curves of martensite ) from the area of partial transformation during heating compressive stresses increase, but then due to austenitization transform into tensile.During transformation the cooling begins, what causes further, much slower growth of tensile stresses.Transformations during cooling cause the decrease in values of tensile stresses and change of their sign.

Figure 22 .
Figure 22.Normal stresses at point 1 during surfacing

Figure 25 .
Figure 25.Normal stresses at point 4 during surfacing